an introduction to the representation theory of temperley ...the primary goal of this thesis is to...

An introduction to the representation theory ofTemperley-Lieb algebras

Jim de Groot

Spring 2015

Bachelorscriptie

Supervisor: prof. dr. J. V. Stokman

Korteweg-de Vries Instituut voor WiskundeFaculteit der Natuurwetenschappen, Wiskunde en Informatica

Universiteit van Amsterdam

Abstract

This thesis starts with the definition of the Temperley-Lieb algebra and two affine analogs,which depend on n ∈ N and q ∈ C×. The Temperley-Lieb algebra is proven to be isomorphic toa diagram algebra.

An overview of Temperley-Lieb modules and intertwining operators is given. It is proventhat the algebra is semisimple for generic q and in the non-semisimple case its principle inde-composable modules are constructed. Thereafter, the connection to statistical physical modelsis described and some of the representations are decomposed in irreducibles or indecompos-ables.

Finally, affine Temperley-Lieb modules and intertwiners are given. In particular, the affinedimer representation is defined and is linked to the link state-modules via an intertwiningoperator.

Title: Representations of the Temperley-Lieb algebraAuthor: Jim de Groot, [email protected], 6265898

Supervisor: prof. dr. J. V. StokmanSecond grader: prof. dr. E. M. OpdamDate: Spring 2015

Korteweg-de Vries Instituut voor WiskundeUniversiteit van AmsterdamScience Park 904, 1098 XH Amsterdamhttp://www.science.uva.nl/math

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http://www.science.uva.nl/math

Contents

1 Introduction 5

2 The Temperley-Lieb algebra 72.1 The Temperley-Lieb algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Diagrams and link states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Relation to the Hecke algebra and braid group . . . . . . . . . . . . . . . . 11

2.1.3 A central element of TLn(β) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2 The extended and reduced affine Temperley-Lieb algebra . . . . . . . . . . . . . . 15

3 Representations of the Temperley-Lieb algebra 173.1 Link state-modules and identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.1 Standard modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.2 Restricted modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.1.3 Induced modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Spin chain representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 The spin representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 First link-spin intertwiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2.3 Second link-spin intertwiner . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 The dimer representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.4 Overview of TLn(β)-modules and their connections . . . . . . . . . . . . . . . . . . 37

4 The structure of TLn(β) 394.1 Semisimplicity for generic q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.1 The radical of a standard module . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.1.2 Gram matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 The cases q = ±i and q = ±1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.1 Bratelli diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.3.2 Irreducibility of the radicals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.3 Principal indecomposable modules . . . . . . . . . . . . . . . . . . . . . . . . 58

5 Examples in statistical physics 635.1 The spin chain model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5.1.1 The Heisenberg XXZ spin- 12

chain model . . . . . . . . . . . . . . . . . . . . 63

5.1.2 Conjecture about the structure of (C2)⊗n . . . . . . . . . . . . . . . . . . . . 64

5.2 The dimer representation revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5.2.1 Connection to the dimer model . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.2 The structure of the Dimer representation . . . . . . . . . . . . . . . . . . . . 66

5.3 Fully and completely packed loop model . . . . . . . . . . . . . . . . . . . . . . . . . 67

6 Representations of the affine Temperley-Lieb algebra 696.1 Link state-modules of aTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1.1 The matchmaker representation . . . . . . . . . . . . . . . . . . . . . . . . . . 69

6.1.2 The singles representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.2 Spin representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2.1 Simple spin representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.2.2 Reduced spin representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

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4 Contents

6.3 Affine dimer representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Conclusion 83

8 Populaire samenvatting 85

Bibliography 89

A Miscellaneous 91A.1 The central element Jn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

A.2 Preliminary representation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Chapter 1

Introduction

The Temperley-Lieb algebra was first introduced by Neville Temperley and Elliott Lieb in 1971

[28]. The family of algebras plays an important role throughout mathematics and physics, asit underlies the study of Potts models, ice-type models and the Andrews-Baxter-Forrestermodels. Moreover, the Temperley-Lieb algebra can be connected to Categorical QuantumMechanics and even to Logic and Computation [1].

The primary goal of this thesis is to give a thorough introduction to the representation the-ory of Temperley-Lieb algebras. Although some papers present an overview about a subjectconcerning the Temperley-Lieb algebra (for example the structure of the algebra in [26]), mostknowledge is scattered around different papers. The main results of the thesis are:

• describing when the Temperley-Lieb algebra is semisimple and describing its structurein terms of irreducible and principal indecomposable modules when it is not in theorem4.13, corollary 4.20 and theorem 4.32.

• defining a new affine Temperley-Lieb representation that is connected to the dimermodel and giving a connection to the well know standard modules in lemma 6.15 andproposition 6.16.

Let us go through the chapters one by one. This thesis starts with two equivalent definitionsof the Temperley-Lieb algebra. First as a diagram algebra and second as an algebra withgenerators subject to defining relations. The definitions are proven to be equivalent and theconnection of the Temperley-Lieb algebra to the Hecke algebra and the braid group is brieflynoted. Besides, a new central element Jn is constructed, which will replace a known centralelement Fn in existing proofs. The second section generalises the definition of the Temperley-Lieb algebra to establish the affine Temperley-Lieb algebra.

In chapter 3 an overview of Temperley-Lieb representations is given, starting with linkstate-modules, which can be viewed as diagrams and are therefore very intuitive. Thereaftertwo representations on C2 ⊗ ⋯ ⊗ C2 (n copies) are analysed: the spin chain module and thedimer representation. Apart from defining them, the representations are linked to each othervia homomorphisms.

Chapter 4 studies the structure of the Temperley-Lieb algebra, which often appears to besemisimple. In the non-semisimple case, complete sets of irreducible modules and principleindecomposable modules are constructed.

In order to justify the study of this algebra, chapter 5 shows examples of its usefulness instatistical physics. It concisely describes the spin chain model and the dimer model, bothwell-known theoretical physical models.

Although our main focus lies on the Temperley-Lieb algebra we do discuss some importantrepresentations of the affine Temperley-Lieb algebra in chapter 6. Both the link state- and thespin chain-modules are defined for the affine Temperley-Lieb algebra, and many homomor-phisms of representations are modified to work for affine Temperley-Lieb representations. Inparticular, the third section generalises the dimer representation to the affine Temperley-Liebalgebra and gives an intertwiner between the link state-modules and the dimer representa-tion. To our best knowledge, both the affine dimer representation and intertwiner have notappeared in literature before.

The thesis concludes with a conclusion and recommendations for further research, and apopular summary in dutch.

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6 Chapter 1. Introduction

Preliminaries

In order to read the thesis, basic knowledge about algebra and representation theory is as-sumed. An undergraduate course on both subjects should suffice. Some more advancedrepresentation theoretic results we use are stated in appendix A.2. Apart from these prelimi-naries, the thesis is largely self-containing.

Notation

The (affine) Temperley-Lieb algebra depends on two or three parameters, the “size” n ∈ Nand complex numbers β and α, and will be denoted by TLn(β). Two versions of the affineTemperley-Lieb algebra discussed in the text are denoted by aTLn(β) and rTLn(β,α). Ele-ments of the Temperley-Lieb algebra are denoted by x, y and z. Representation homomor-phism are usually denoted by lower case greek letters higher than κ, e.g. ρ, ζ, µ and elementsof representations are called u, v and w. For homomorphisms of representations capital greekletters are reserved, in particular Ψ,Ω and Γ will be important homomorphisms.

When extending a Temperley-Lieb representation or intertwiner to the affine Temperley-Lieb algebra, the notation is ofter preserved and marked with a tilde. For example ζ and µ

are affine Temperley-Lieb representations and Ω and Γ are homomorphisms between affineTemperley-Lieb representations.

Acknowledgments

I would like to thank my supervisor Jasper Stokman for introducing me to the subject andgranting me a generous amount of his time. Answering most of my questions and sometimessaying “I don’t know, go figure it out,” has helped this thesis take its present form.

Chapter 2

The Temperley-Lieb algebra

The Temperley-Lieb algebra was first introduced by Neville Temperley and Elliott Lieb in 1971

[28]. The family of algebras plays an important role throughout mathematics and physics, as itunderlies the study of Potts models, ice-type models and the Andrews-Baxter-Forrester mod-els. Since its introduction it has been a subject of interest and still new research is published(most recently, in 2015, a paper on the connection of a Temperley-Lieb representation with thedimer model was published).

This chapter lays the foundation of the study of the representation theory of the Temperley-Lieb algebra. Two types of families are defined. The first section treats the “normal” Temperley-Lieb algebra, which depends on a positive integer n a parameter β ∈ C, defining it in severalequivalent ways and examining some of its elements. The second section discusses the affineTemperley-Lieb algebra, which depends on n ∈ Z≥1, the parameter β and a second parameterα. It extends the definition from the first section.

2.1 The Temperley-Lieb algebra

In the first subsection we will define the Temperley-Lieb algebra in two equivalent ways: asa diagram algebra and as an algebra with n − 1 generators satisfying defining relations. Thesubsequent subsection studies the relation of the Temperley-Lieb algebra to the Hecke algebraand the braid group. The Temperley-Lieb algebra turns out to be a quotient of both the Heckealgebra and the group algebra of the braid group. Finally, in subsection 2.1.3, a central elementof the algebra is constructed. It helps getting used to the algebra and appears useful later on.

2.1.1. Diagrams and link states

We commence by defining an n-diagram and the diagram algebra.

2.1 Definitions. An n-diagram consists of two parallel lines with n vertices on both lines.These vertices are numbered from top to bottom by 1, . . . , n on the left and 1, . . . , n on theright and the vertices i and i lie on the same height. The vertices must be connected byedges such that the edges lie in between the two parallel lines, do not cross one another andeach vertex is the endpoint of exactly one edge. We call an edge between two vertices a linkand we say that two vertices i, j are connected if there is a link from i to j. We call a linkquasi-simple if it connects two vertices on the same line and simple in i if it connects the i-thand the (i + 1)-th vertex on the same line. Finally, call a link from vertex i to i straight. Iftwo diagrams give the same pairing of the set 1, . . . n, 1, . . . , n, we view them as the samediagram.

Let `n denote the set of all n-diagrams and let β ∈ C be a complex number. Define thediagram algebra to be the formal vector space with a basis of n-diagrams over C, denotedC`n(β) (thus elements of C`n(β) are linear combinations of n-diagrams). Define the productof two elements in `n to be the concatenation of the diagrams. If a circle forms, remove itand multiply the result in C`n(β) by a factor β. Extending this construction linearly inducesa bilinear product on C`n(β), making it an algebra. Clearly this product is associative. Theelement with only straight links (see the left diagram in figure 2.1) acts as the identity with

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8 Chapter 2. The Temperley-Lieb algebra

respect to the product (accordingly we will call it 1), hence C`n(β) is in fact an associativering with unit.

Denote by ei the diagram which has simple links in i and i and has straight links on allother vertices.

We may write β = q + q−1 with q ∈ C. This will appear to be useful later on. If no ambiguitycan occur we will drop the β and write C`n for C`n(β). Note that the defined operation neednot be commutative or invertible. To get used to the definitions, let us have a look at someexamples.

Examples. Figure 2.1 depicts four elements of `4. The most left element in the figure is theidentity in C`4. The second diagrams has simple links in 1 and 2. The third element is e3. Theright element is the product of the previous three.

1

2

3

4

1

2

3

4

Figure 2.1: The elements 1, e1 ⋅ e2, e3 and their product in C`4.

Set e = e1e2 in C`4. Then ee2 = β ⋅ e, see figure 2.2.

× = β ⋅ .

Figure 2.2: Multiplication causing a loop.

One can easily find some identities with respect to this multiplication. For example e2i = β ⋅ei

and eiei±1ei = ei. Also, for ∣i − j∣ ≥ 2 the elements ei and ej commute. These relations can beseen by drawing the corresponding diagrams.

It appears that the set `n is generated by 1, e1, . . . , en−1.

2.2 Lemma. The set e1, . . . , en−1 generates the diagram algebra C`n as a complex associative algebra.

Proof. It suffices to show that the generators can make links within links and links that moveup or down. The first one can be achieved by taking eiei−1ei+1 (draw this), the second one byeiei±1. A concise proof can be found in [18].

2.3 Definition. An (n, p)-link state is obtained by cutting an n-diagram in half and lookingat the left half. Some vertices may still be connected, but there might also be some looseends, called defects. Still, the edges cannot cross. Note that a defect cannot occur in betweentwo connected vertices. In an (n, p)-link state, n is the number of vertices and p denotes thenumber of quasi-simple links on the left line.

We can identify each (n, p)-link states with an increasing path in Z2 starting in (0,0) andending in (n−p, p) by going through the link state from top to bottom and taking one step upin Z2 whenever a link is closed and one to the right otherwise.

2.1. The Temperley-Lieb algebra 9

Figure 2.3: From left to right, a (4,1)-, (4,2)- and (5,1)-link state.

Figure 2.4: Paths in Z2 corresponding to the link states in figure 2.3.

The set of (n, p)-link states corresponds 1-1 with the set of inceasing paths (0,0)→ (n−p, p)that lie (non-strictly) beneath the diagonal. The number of such paths is computed in thefollowing lemma.

2.4 Lemma. The number of increasing paths in Z2 starting in (0,0) and ending in (n − p, p) that lienon-strictly beneath the diagonal equals

(np) − ( n

p − 1).

Proof. An arbitrary increasing path from (0,0) to (n−p, p) has length n and is fixed by choosingin which steps it goes up, hence the total number of such paths is (n

p).

If a path does not lie beneath the diagonal, it must touch the line y = x+1. Let P be the firstpoint where that happens, and reflect the part of the path from (0,0) to P around the liney = x + 1. The result is a path from (−1,1) to (n − p, p). Conversely, given a path from (−1,1)to (n − p, p) we can reflect the part of the path from (−1,1) to the first point where it touchesthe line y = x + 1 around this line to obtain a path from (0,0) to (n − p, p) which crosses thediagonal or lies above it. (Note that it must cross this line, since (−1,1) lies at the left of it,whereas (n − p, p) is on the right of the line.) This gives a bijection between paths that do notlie underneath the diagonal and paths from (−1,1) to (n − p, p). The number of such pathsequals ( n

p−1). This proves the lemma.

Set dn,p = (np) − ( n

p−1). It is easily verified that

dn,p = dn−1,p + dn−1,p−1. (2.1)

This equality will be useful later on. The number dn,p emerges in the following lemma.

2.5 Proposition. The number of (n, p)-link states is dn,p = (np) − ( n

p−1). Thereto #`n = d2n,n =

(2nn) − ( 2n

n−1) and dimC`n = d2n,n.

Proof. The first statement we have already seen. As for the second one, note that n-diagramsare in bijection with the (2n,n)-link states by rotating the right line 180 degrees clockwise andplacing it under the left line (see figure 2.5). The third statement is an immediate consequenceof the second one.


Figure 2.5: Bijection between n-diagrams and (2n,n)-link states.

We will now define the Temperley-Lieb algebra. Proposition 2.9 shows that it is isomorphicto the diagram algebra C`n(β).

2.6 Definition. The Temperley-Lieb algebra, denoted by TLn(β), is the associatve unitalalgebra over C generated by the elements e1, . . . , en−1 satisfying the defining relations

e2i = β ⋅ ei, eiei±1ei = ei and eiej = ejei if ∣i − j∣ ≥ 2. (2.2)

A word in TLn(β) is a product of generators. Call a word reduced if it cannot be shortenedusing the relations from (2.2).

If no confusion can occur, we write TLn instead of TLn(β).The algebra TLn is an associative algebra with unit 1. Obviously, if 1 occurs in a word

it may be omitted. We will now prove some facts about these words and define the Jones’normal form (in proposition 2.8).

2.7 Lemma. In a reduced word ei1⋯eik , the maximal index m = maxi1, . . . , ik occurs only once.

Proof. We prove this by induction. Suppose a word is reduced and em occurs twice or more.Then a part of the word looks like ⋯emEem⋯, where the maximal index in E is smaller thanm. Since the whole word is reduced, so is E, hence by assumption its maximal index m′

occurs only once.If m′ <m − 1 then em and E commute, so that we can write ⋯Ee2

m⋯ and cut out the square(using (2.2)). If m′ =m−1 then all elements except one occurence of em′ commute with em andwe can move the em’s to the left and right of em′ to find emem′em = em. Both cases contradictthe assumption that the word is reduced.

2.8 Proposition. Let E be a reduced word in TLn. Then it can be written as a sequence of decreasingsequences of generators (called the Jones’ normal form)

E = (ej1ej1−1ej1−2⋯et1)(ej2ej2−1ej2−2⋯et2)⋯(ejkejk−1ejk−2⋯etk)

so that 0 < j1 < j2 < ⋯ < jk < n and 0 < t1 < t2 < ⋯ < tk < n. Besides, any reduced word may also bewritten as

E = (ej1ej1+1ej1+2⋯et1)(ej2ej2+1ej2+2⋯et2)⋯(ejkejk+1ejk+2⋯etk)with n > j1 > j2 > ⋯ > jk > 0 and n > k1 > k2 > ⋯ > kt > 0 (called the reverse Jones’ normal form).

Proof. Let E be a reduced word and let m be the unique maximal index (see previous lemma).Move em to the right of the word as far as possible using only the third relation from definition2.6 until it is either all the way to the right, or the letter next to it is em−1. Now move emem−1

to the right until it is all the way to the right, or the next letter is em−2. Repeat the process.


Then the result is of the form E = E′ ⋅ (emem−1em−2⋯ek). In addition, the word is still reducedand E′ is a reduced subword of E with maximal index less than m. Repeating the processinductively yields the desired form and j1 < j2 < ⋯ < jk.

Assume ti ≥ ti+1, then ti = ji+1 − s for some s and we have

E = ⋯(ejieji−1⋯eti+1eti)(eji+1eji+1−1⋯eji+1−s+1eji+1−s⋯eti+1)⋯= ⋯(ejieji−1⋯eti+1)(eji+1eji+1−1⋯ etieji+1−s+1eji+1−s

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=eji+1−s

⋯eti+1)⋯

which shows that E was not reduced. Contradiction. Hence 0 < t1 < t2 < ⋯ < tk < n, provingthe first statement.

The second statement can be proven in a similar fashion.

Using the Jones’ normal form of a reduced word one can identify reduced words in TLnwith walks from (0,0) to (n,n) in Z2 that do not cross the diagonal. This is done as follows,let E = (ej1ej1−1ej1−2⋯etk) be a reduced word, then the walk is

(0,0)→ (j1,0)→ (j1, t1)→ (j2, t1)→ (j2, t2)→ ⋯⋯→ (jk, tk−1)→ (jk, tk)→ (n, tk)→ (n,n).

The path cannot cross the diagonal because ji+1 > ji ≥ ti for 1 ≤ i ≤ k.Different reduced words produce different paths and from each path we can construct a

word in Jones’ normal form. The number of different reduced words is bounded from aboveby the number of paths. This leads to the equivalence of C`n and TLn.

2.9 Proposition. The diagram algebra C`n(β) and the Temperley-Lieb algebra TLn(β) are isomorphicfor fixed n ∈ N and β ∈ C via the isomorphism

TLn(β)→ C`n(β) ∶ ei ↦ i + 1i

.

Proof. We have seen that C`n satisfies the relations from TLn, so it is a quotient of TLn.Therefore

(2n

n) − ( 2n

n − 1) ≥ dim TLn ≥ dimC`n = (2n

n) − ( 2n

n − 1),

hence dim TLn = dimC`n and the latter cannot satisfy any other relations over C, ultimatelyproving the isomorphism.

The result allows us to identify the diagram algebra and the Temperley-Lieb algebra, hencewe will use both definitions interchangeably.

2.1.2. Relation to the Hecke algebra and braid group

It can be useful to view the Temperley-Lieb algebra as a quotient of the Hecke algebra, whichis in turn a quotient of the group algebra of the braid group. (For a definition of the groupalgebra, see example 2.2.4 of [7]).

The braid group was first introduced explicitly by Emil Artin in 1925. Elements of the braidgroup can be represented by diagrams much like the Temperley-Lieb algebra. The theory ofbraid groups is well developed and is used in e.g. knot theory. For more on braid groups seefor example [16] by Kassel and Turaev. We will be content with only the definition.


2.10 Definition. The braid group Bn is the group generated by generators σ1, σ2, . . . , σn−1

satisfying the braid relations σiσi+1σi = σi+1σiσi+1 and σiσj = σjσi when ∣i − j∣ ≥ 2.

The first braid group B1 is trivial. The second braid group B2 is generated by a singlegenerator and no relations, thus B2 ≅ Z. For n ≥ 3, Bn is a non-abelian infinite group.

Let us now define the Hecke algebra, named after German mathematician Erich Hecke, andshow that it is a quotient of the group algebra of the braid group. (Besides our purpose touse this connection to study the Temperley-Lieb algebra, it has a remarkable application inthe construction of new invariant knots, see for example [20].)

2.11 Definition. Let q ∈ C×. The Hecke algebra Hn(q) is the associative algebra over Cgenerated by T1, . . . , Tn−1 with defining relations the braid relations TiTi+1Ti = Ti+1TiTi+1,TiTj = TjTi when ∣i − j∣ ≥ 2, and the quadratic relation

(Ti − q)(Ti + q−1) = 0.

2.12 Proposition. Let CBn denote the group algebra of the braid group. There exists a unique surjec-tive homomorphism of algebras (cf. definition A.2) φ ∶ CBn →Hn(q) such that

σi ↦ q−1/2Ti

for 1 ≤ i ≤ n − 1.

Proof. Define a map φ from the set σ1, . . . , σn−1 into the group Hn(q)∗ by φ(σi) = q−1/2Ti for1 ≤ i ≤ n − 1. The map induces a group homomorphism from the free algebra Z ∗ . . . ∗ Z intoHn(q)∗. Since

φ(ei)φ(ej) = q−1TiTj = q−1TjTi = φ(σj)φ(σi)

when ∣i − j∣ ≥ 2, and

φ(σi)φ(σi+1)φ(σi) = q−3/2TiTi+1Ti = q−3/2Ti+1TiTi+1 = φ(σi+1)φ(σi)φ(σi+1).

the ideals corresponding to the braid relations lie in the kernel of φ. Hence φ induces ahomomorphism φ ∶ Bn → Hn(q)∗, which in turn may be linearly extended to all of CBn toacquire the algebra homomorphism

φ ∶ CBn →Hn(q).

Surjectivity follows from the fact that each generator Ti of Hn(q) is the image of q1/2σi ∈CBn.

The previous proposition implies Hn(q) is isomorphic to the quotient of CBn with thetwo-sided ideal ker φ. Similarly,

2.13 Proposition. The map h ∶Hn(q)→ TLn(β) given by Ti ↦ ei − q−1 is a unit preserving algebrahomomorphism.

Proof. Again, it suffices to show that the image under h of an element in Hn(q) does notdepend on the chosen representation of the element. We need to show that h(Ti) satisfies thedefining relations of Hn in terms of Ti.


First, calculate

h(Ti)h(Ti+1)h(Ti) = (ei − q−1)(ei+1 − q−1)(ei − q−1)= eiei+1ei − q−1(eiei+1 + e2

i + ei+1ei) + q−2(ei + ei+1 + ei) − q−3

= ei − q−1(eiei+1 + (q + q−1)ei + ei+1ei) + q−2(ei + ei+1 + ei) − q−3

= ei − q−1qei − q−2ei − q−1(eiei+1 + ei+1ei) + q−2(ei + ei+1 + ei) − q−3

= −q−1(eiei+1 + ei+1ei) + q−2(ei + ei+1) − q−3

Similarly h(Ti+1)h(Ti)h(Ti+1) = −k−1(eiei+1 + ei+1ei) + k−2(ei + ei+1) − k−3, so the first relationfrom definition 2.11 is satisfied. Second, h(Ti)h(Tj) = (ei − q−1)(ej − q−1) = h(Tj)h(Ti) for∣i − j∣ ≥ 2 and finally compute (h(Ti) − q)(h(Ti) + q−1) = (ei − q − q−1)ei = e2

i − βei = 0.The element Ti + q−1 ∈ Hn(q) is mapped under h to ei, hence the generators of TLn are in

the image of h implying the surjectivity.

We now have algebra homomorphisms

CBn Hn(q) TLn(β).φ h

It follows that TLn(β) =Hn(q)/kerh. Moreover, TLn(β) can be obtained as a quotient CBn/Iof the group algebra of the braid group by taking I = φ−1(kerh) ⊂ CBn.

2.1.3. A central element of TLn(β)

In this subsection we construct a central element of TLn, that is, an element that commuteswith all other elements of TLn. In general, central elements can be used for determining thestructure of a representation. The action of a central element commutes with the action of thealgebra on the representation space and decomposes the representation space in generalisedeigenspaces. These generalised eigenspaces are usually smaller and easier to study. In thisthesis, a central element will be used in the study of the structure of TLn in chapter 4.

There are many known central elements of the Temperley-Lieb algebra. One of them is Cn,which is derived from its analog in the braid group and is defined by

Cn = ((1 − qe1)(1 − qe2)⋯(1 − qen−1))n ∈ TLn

(cf. [3],[5]).We define a central element Jn which is inspired on the element Fn defined by Ridout

and Saint-Aubin in [26]. Our element differs in that the diagrams used to define it admit thesecond and third Reidermeister moves.

2.14 Definition. Let β = q+q−1 and let b ∈ C× be such that b2 = q. Set c = bi, where i ∈ C denotesthe imaginary element. Write = c + c−1 and = c + c−1 . Define Jn as infigure 2.6.

This has to be read as the sum over all possible tilings of the crossings (multiplied bythe given factor). This sums over 22n diagrams and each of the summands is indeed a n-diagram. The following example gives an explicit computation of the element J1 in terms ofTL-diagrams.


Jn =

Figure 2.6: The central element Jn.

Example. The element J1 is computed as follows,

J1 =

= c2 + + + c−2

= −q(q + q−1)1 + 1 + 1 − q−1(q + q−1)1= −(q2 + q−2)1.

In a similar way, one can attain J2, J3

J2 = (q3 + q−3)1 + (2 − (q2 + q−2))e1

J3 = (−2 + 3β + β2 − β4)1 + (4β − 3β3)(e1 + e2) + (4 − β2)(e1e2 + e2e1).

Although the definitions of Fn and Jn look very similar, the resulting elements differ. Forexample, compare J2 and J3 with F2 and F3 below,

F2 = (q3 + q−3)1 − (q − q−1)2e1,

F3 = (q4 + q−4)1 − (q − q−1)(q2 − q−2)(e1 + e2) + (q − q−1)2(e1e2 + e2e1).

2.15 Proposition. The element Jn ∈ TLn is central, i.e. xJn = Jnx for all x ∈ TLn.

Proof. It suffices to prove eiJn = Jnei for all generators ei of TLn. We look only at the levelwhere ei acts on Jn. First, note that

= −q + + − q−1 = ,

and likewise

= , = and = .

Now it follows that

= = =

2.2. The extended and reduced affine Temperley-Lieb algebra 15

and

= = = .

We see that eiJn = Jnei, which proves the proposition.

For the reader who has some knowledge of knot theory, the following corollary is interest-ing. (For the reader who wishes to get familiar with knot theory, we refer to [20].)

2.16 Corollary. The crossings used to construct Jn satisfy the second an third Reidemeister moves.

2.2 The extended and reduced affine Temperley-Lieb algebra

In the extended affine Temperley-Lieb algebra we wrap the diagrams around a cylinder. Con-sequently, n may be connected to 1 by a simple link. This allows us to have a link from n to1 without crossing any link in the rectangle between (0, n − 1) and (1,2). We need an extragenerator en, the diagram with a link from 1 to n (see figure 2.7). Also we need a generator uwhich sends j to j + 1 mod n for all vertices j and a generator u−1 which acts as the inverseof u and sends j to j − 1 mod n.

2.17 Definition. Let β ∈ C. The extended affine Temperley-Lieb algebra over C, denotedaTLn(β), is the associative C-algebra with unit given by the generators, u, u−1 and ei, i ∈ Z/nZand defining relations

(i) e2i = βei,

(ii) eiej = ejei when i ≠ j ± 1 mod n,(iii) eiei±1ei = ei,(iv) uei = ei−1u,(v) uu−1 = 1 = u−1u,

Here, u denotes the affine diagram that connects vertex i on the left to i + 1 on the right (cffigure 2.8).

Remark. In the previous section we proved that the diagram algebra and the Temperley-Liebalgebra coincide, wherefore we could use them interchangeably. It would be decent to do thesame for the algebra we defined in this section. An analogous result holds when identifyingid, ei, u and u−1 with the obvious diagrams. For a proof, however, we refer to [14]. As withTLn, we will use the two definitions interchangeably.

The multiplication of diagrams corresponding to the affine Temperley-Lieb algebra is sim-ilar to that of TLn(β), by placing the cylinder diagrams next to each other and following thelinks. Contractible loops may be removed by multiplying with a factor β.

Note that there are infinitely possible diagrams, whereas there are only finitely many di-agrams in the sense of definition 2.1, since a link can make an arbitrary number of loopsaround the cylinder before reaching its destination.

Similar to the TLn-algebra from the previous section, we will be allowed to delete con-tractible loops by multiplying with a factor β. But on a cylinder, as the following exampleshows, we may have loops around the cylinder. These are not contractible. We will introducea factor α ∈ C× by which we can multiply in order to delete these loops. This will reduce thealgebra aTLn, thereupon it is called the reduced affine Temperley-Lieb algebra. First, we givean example of a non contractible loop.


Figure 2.7: The n-th generator en.

Figure 2.8: The element u.

× = = α ⋅ .

Figure 2.9: Multiplication causing a non-contractible loop.

Example. Figure 2.9 shows how a non-contractible loop arrises and is deleted.

2.18 Definition. Let α ∈ C×. The reduced affine Temperley-Lieb algebra, rTLn(β,α) is thequotient of aTL with the two-sided ideal R generated by (eevu±1eev−αeev, eoddu±1eodd−αeodd).Here, eev = e2e4⋯en and eodd = e1e3⋯en−1. Note that eev and eodd are only defined for even n,thus for odd n the ideal will be 0. In short,

rTLn(β,α) = aTLn(β)/R.

This is equivalent to saying rTLn is the associative algebra generated by e1, . . . , en, u, u−1 sat-

isfying all relations from definition 2.17 and a sixth relation,

(vi) eevu±1eev = αeev and eoddu±1eodd = αeodd when n is even.

Remark. One might notice that the affine Temperley-Lieb algebra could be defined with lessgenerators. For example, we could use the generators e1, u and u−1 to define the other genera-tors, since u−1e1u = e2 and in general u−ie1u

i = ei+1. However, in order to do this the definingrelations get much uglier. For example we should have e1u

ie1u−1 = uie1u

−1e1 for 2 ≤ i ≤ n − 1.For simplicity’s sake we have chosen the elaborate set of generators.

If no confusion can occur, we will write aTLn and rTLn instead of aTLn(β) and rTLn(β,α).

Chapter 3

Representations of the Temperley-Lieb algebra

In this chapter we study representations of the Temperley-Lieb algebra. After defining a TLn-module, it is connected to others via intertwiners (homomorphisms of representations). Somepreliminary results in representation theory can be found in appendix A.2.

We start with representations based on link states as defined in definition 2.3. Next, weconsider (C2)⊗n as a TLn-module, which is more commonly used in physics (as will becomeclear in chapter 5). In section 3.3 the dimer representation is introduced, which is connectedto the dimer model in section 5.2. The chapter closes with an overview of the defined repre-sentations and intertwining operators.

3.1 Link state-modules and identities

This section treats the most intuitive TLn-representations. These can be viewed as diagramsand are called link state-modules. The restricted and induced link state-module are intro-duced and connected to “normal” link state-modules via short exact sequences.

3.1.1. Standard modules

Recall concatenation of n-diagrams of the Temperley-Lieb algebra. Using this idea, define theaction of a n-diagram x in TLn with an (n, p)-link state v in a similar way. When Mn denotesthe complex span of (n, p)-link states (for fixed n and arbitrary p ∈ 0, . . . , ⌊n/2⌋), let µ(x)vbe the concatenation of x and v. Following the links starting on the left gives a new link state.Loops may be removed by multiplying with a factor β. Any other line segments (that arenot loops and not connected to a vertex on the left line) we delete. Linearly extending thisdefinition yields a map

µ(x) ∶Mn →Mn ∶ v ↦ µ(x)vfor all x ∈ TLn and v ∈ Mn. One can easily see that µ(1) = id, the identity map on Mn, andµ(xy) = µ(x)µ(y). Hence Mn is a TLn-module.

Example. In M4 we can compute the action of a certain v ∈M4 with e2 ∈ TL4 as follows,

⋅ = = .

Note that the action of x ∈ TLn may close two defects creating an extra link, but can neverproduce extra defects. Hence the complex span of link states with at least p links is a sub-module of Mn. Furthermore it makes sense to take quotients as in the following definition.

3.1 Definitions. (i) The complex span of all (n, p)-link states (for fixed n and arbitrary p) isa representation of TLn. It is denoted Mn and called the link module.

(ii) The complex span of all (n, p′)-link states with p′ ≥ p is denoted by Mn,p and is a sub-module of Mn.

17

18 Chapter 3. Representations of the Temperley-Lieb algebra

(iii) The quotient of Mn,p and Mn,p+1 is generated by the equivalence classes of (n, p)-linkstates (with p fixed) and is written by

Vn,p ∶=Mn,p

Mn,p+1.

The Vn,p (0 ≤ p ≤ n/2) are called the standard modules of TLn(β). We denote the set oflink states with precisely p links by Bn,p and write Bn,p for the set of equivalence classesof elements in Bn,p in Vn,p.

Denote the representation map of TLn on Vn,p by µn,p. Note that all of the above algebrashave a canonical basis of (n, p)-link states. In particular, Bn,p is a basis for Vn,p. From now on,when dealing with link states in Vn,p we will neglect to state that they are actually equivalenceclasses (yet nevertheless keeping it in mind). For example, figure 3.1 gives a basis for V5,2.

Figure 3.1: Basis Bn,p for V5,2.

3.2 Remark. We often define a homomorphism ρ corresponding to a representation V bymapping the generators e1, . . . , en−1 of TLn into EndV . If x ∈ TLn we can write x = ei1ei2⋯eimand set ρ(x) = ρ(ei1)ρ(ei2)⋯ρ(eim). This automatically turns ρ into an algebra homomor-phism, provided that the image of x is not dependent on the choice of notation of x. Henceto proof that ρ is a homomorphism, we have to check that

ρ(ei)2 = β ⋅ ρ(ei),ρ(ei)ρ(ei±1)ρ(ei) = ρ(ei) andρ(ei)ρ(ej) = ρ(ej)ρ(ei) if ∣i − j∣ ≥ 2.

Using this insight, one can easily verify that the modules in definition 3.1 are indeed repre-sentations of TLn(β).

We will now define the composite module.

3.3 Lemma. Set Wn,p = span(Bn,p ∪ Bn,p−1) as a set. Then Wn,p is a TLn(β)-module via

θp ∶ TLn(β)→ End(Wn,p),

where θp is defined as follows. On a link state in Bn,p, the action of TLn is the same as on itsequivalence class in Vn,p. The action of x ∈ TLn(β) on w ∈ Bn,p−1 is separated into three cases:

(i) If no defects are closed by composition, θp(x)w = µ(x)w.(ii) If one extra link occurs (i.e. two defects are connected) and one of the connected defects is the

lowest defect (that is, the defect with the highest numbered vertex), then θ(x)w is obtained byidentifying the composition with the corresponding link state in Bn,p and multiplying by β foreach closed loop.

3.1. Link state-modules and identities 19

(iii) Otherwise, if two defects are connected and none of them is the last defect, or more that two defectsare connected (more than two extra links appear), set θ(x)w = 0.

The map θp is the linear continuation of the generators. This module is called the composite module.

Proof. The proof is straightforward.

One readily sees that Vn,p is a submodule of Wn,p and Wn,p/Vn,p ≅ Vn,p−1, yielding theshort exact sequence

0 Vn,p Wn,p Vn,p−1 0.

Let us now define the first module homomorphism or intertwining operator of this thesis,after lemma 5.4 of [22]. For n ∈ 2N define the element

yn = − + . . . + (−1)n/2−1 ∈ Vn,1.

Note that eiyn = 0 in TLn(0).Define the intertwiner Υp ∶ Vn,p−1 → Vn,p as follows. Given a link state v ∈ Bn,p−1, tem-

porarily erase the p − 1 links, replace the n − 2(p − 1) defects by yn−2(p−1) (thereby producingan extra link) and then put the links back in their original positions. Linearly extending thisconstruction yields the map Υp.

Example. The following example illustrates the action of Υp on a link state in Vn,p−1.

Υp

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

= = − .

3.4 Proposition. Let β = 0 and n even. Then Υp intertwiners the TLn(0)-modules Vn,p−1 and Vn,p.That is,

µn,p−1(x)Υp(v) = Υp(µn,p(x)v) (3.1)

for all x ∈ TLn(0), v ∈ Vn,p−1.

Proof. For x = 1 this is trivial. It suffices to verify the identity (3.1) for x = ei and v ∈ Bn,p−1.The general case will then follow from linearity. Let v ∈ Bn,p−1 and consider three cases,corresponding to the number of defects in position i and i + 1.

Case 0. If both i and i + 1 are occupied by links, the action of ei on v does not affect thedefects. It does not matter if this is done before Υ connects defects or after. The same holdswhen i is connected to i + 1, in which case we get a factor β (either before connecting defectsby Υ or after).


Case 1. If one of i and i + 1 is a defect and the other is occupied by a link, then acting on vby ei makes the defect move and changes the link. It does not matter if this is done before orafter closing defects with Υ. The example below the proof should clarify this.

Case 2. If both i and i + 1 are defects, then µ(ei)Υ(v) = 0 since eiyn = 0. On the other handµ(ei)v = 0 in Vn,p−1 since it closes a defect. Hence µ(ei)Υ(v) = Υ(µ(ei)v). This finishes theproof.

Example. The following is an example to go with case 1 of the proof above.

v = , µ(e4)Υp(v) = µ(e4) = = = Υ(µ(e4)v).

Besides providing a neat example of an intertwining operator, the map Υ will come inhandy in section 5.2, where the structure of the dimer representation is studied.

3.1.2. Restricted modules

Starting from any representation V of TLn, one can construct a TLn−1-module (the restrictedmodule) and TLn+1-module (the induced module). We will do so in definitions 3.5 and 3.7.It will turn out in proposition 3.9 that under a weak condition the restriction of the TLn+1-module Vn+1,p+1 and the induced module of the TLn−1-module Vn−1,p are isomorphic.

3.5 Definition. Let V be a TLn-module with representation map ρ. Consider the inclusioni ∶ TLn−1 TLn which is made by adding a link under the existing (n − 1)-diagram. We canrestrict V to a TLn−1-module by defining the action of x ∈ TLn−1 on v ∈ V by x ⋅ v = ρ(i(x))v.Denote this restriction by V ↓ and call it the restricted module.

The case where V = Vn,p is particularly interesting. We can view Vn,p as a TLn−1-module(and call it Vn,p↓). Note that a basis for Vn,p is also a basis for Vn,p↓. Even if Vn,p is irreducibleas TLn-module, Vn,p↓ need not be irreducible anymore. One can easily see that Vn−1,p is asubmodule of Vn,p ↓. There is a trivial inclusion Vn−1,p Vn,p↓ and we know that Vn−1,p isinvariant under TLn−1. The following proposition shows how these modules occur in a shortexact sequence.

3.6 Proposition. There is a short exact sequence of TLn−1-modules

0 Vn−1,p Vn,p↓ Vn−1,p−1 0.ϕ ψ

This entails that Vn−1,p is a submodule of Vn,p↓ (as we have already seen) and Vn,p↓ /Vn−1,p ≅Vn−1,p−1.

Proof. The inclusion Vn−1,p → Vn,p↓ is defined by adding an extra defect to u ∈ Vn−1,p at thebottom position. This is clearly an injective homomorphism.

The quotient Vn,p↓ /Vn−1,p is a TLn−1-module with a basis of cosets represented by (n, p)-link states with no defect at position n.


We can easily define a map

Ψ ∶ Vn,p↓ /Vn−1,p → Vn−1,p−1

by cutting the link to position n (thereby creating two defects) and removing the n-th vertexwith its newborn defect. It is immediate that Ψ is surjective. We claim that Ψ is an intertwiningoperator. Let z be a basis element of Vn,p↓ /Vn−1,p. Suppose that the link to n starts in m. Ifposition m − 1 is not a defect, we have

Ψ(eiz) = eiΨ(z) (3.2)

for 1 ≤ i ≤ n − 1. Since the link from m to n does not disappear by the action of ei, we simplyobtain another basis element. If i = m − 1 and m − 1 is a defect in z, we have em−1z = 0 inVn,p↓ /Vn−1,p for it has a defect at position n, thus Ψ(em−1z) = Ψ(0) = 0. But Ψ(z) has a defectat both m − 1 and m, so concatenation with em−1 will create an extra link, thence em−1Ψ(z)has p links, making it is 0 in Vn−1,p−1. So equation (3.2) always holds, proving that Ψ is anintertwiner. This also proves that ψ ∶ z → Ψ([z]) is a homomorphism with ker(ψ) = ϕ(Vn−1,p)and that the above is indeed an exact sequence of TLn−1-modules.

3.1.3. Induced modules

In this subsection we construct a TLn+1-module from a TLn-module, called the induced mod-ule. The construction is given for a standard module Vn,p, but can be generalised to anyTLn-module.

3.7 Definition. The induced module Vn,p↑ of the TLn-module Vn,p is

Vn,p↑ ∶= TLn+1⊗TLnVn,p.

The action of TLn+1 is given by x(y⊗v) = (xy)⊗v for all x, y ∈ TLn+1 and v ∈ Vn,p. The “TLn”in the subscript of the tensor means that xy⊗ v = x⊗ yv for all x ∈ TLn+1, y ∈ TLn and v ∈ Vn,p,where on the left-hand-side we view y as an element of TLn+1 by adding a horizontal edge atn + 1. So elements from TLn behave as scalars in the tensor product.

An alternative way to define the induced module is by taking the quotient of TLn+1⊗CVn,pwith the module generated by xy ⊗ v − x⊗ yv (x ∈ TLn+1, y ∈ TLn, v ∈ Vn,p).

Using the Jones’ normal form (propostion 2.8) we can see that TLn+1 is spanned by the setEerer+1⋯en and E′, with E,E′ words in TLn. Let B be a basis for Vn,p, then

1⊗ b, en ⊗ b, en−1en ⊗ b, . . . , e1e2⋯en ⊗ b ∣ b ∈ B (3.3)

spans the set Vn,p↑. In general this set does not form a basis for Vn,p↑, as we see the followingexample.

Example. If b can be written as en−1b′ for some b′ ∈ Vn,p, then

en−1en ⊗ b = en−1en ⊗ en−1b′ = en−1enen−1 ⊗ b′ = en−1 ⊗ b′ = 1⊗ en−1b

′ = 1⊗ b.

In the remainder of the subsection, we will investigate how we can restrict the spanningset in (3.3) to make it into a basis, resulting in corollary 3.10. Furthermore we obtain anidentity between restricted and induced modules. Before we commence, we have a look atsome TLn-modules with small n.


Example. When β = 0 we have V2,1 ≃ V2,0 and V2,1↑ ≃ V3,1 ⊕V3,0.

If b = eib′ for some b′ ∈ Vn,p then b has a simple link at i. Conversely, suppose n ≥ 3 andb ∈ Vn,p has a simple link at i, then we can set b′ = ei+1b to find b = eib′ (for i < n− 1, if i = n− 1,set b′ = ei−1b). See also figure 3.2.

b =i + 1

i

i − 1b′ =

i + 1i

i − 1

Figure 3.2: If b is simple at i then b = eib′.

If b ∈ TL2(β) has a simple link in 1 and β ≠ 0 we can set b′ = 1βe1 to find b = e1b

′. Whenβ = 0 such an expression is not possible. Call b ∈ Vn,p r-admissible if it has no simple links ati ≥ r. Every element is n-admissible. The element erer+1⋯en ⊗ b is called r-admissible if b isr-admissible.

3.8 Lemma. Let n ≥ 3, p ≤ ⌊n/2⌋ and let e ∈ TLn+1 be a word in the generators and b ∈ Vn,p.Then there exist s ∈ N with s ≤ n + 1 and b′ ∈ Vn,p that is either 0 or s-admissible such that e ⊗ b =eses+1⋯en ⊗ b′ in Vn,p↑. (When s = n + 1 we get e⊗ b = 1⊗ b′.)

Proof. Write e in reverse Jones’ form (cf. proposition 2.8). If en does not occur in this formthen we have e ⊗ b = 1 ⊗ eb, so we can set s = n + 1 and b′ = eb. Otherwise, we may writee = erer+1⋯ene′, where e′ is a word without en in it. We see that

e⊗ b = erer+1⋯en ⊗ e′b.

If e′b is r-admissible we may set s = r and b′ = e′b to find the desired result.Most trouble occurs when e′b is not r-admissible, that is, e′b has a simple link at i ≥ r.

Suppose i is the highest numbered vertex with a simple link at it (i.e. there is a simple linkfrom i to i+ 1). We have seen that we can write any element e′b as eib′′ when n ≤ 3. Hence wemay write

erer+1⋯en ⊗ e′b = erer+1⋯en ⊗ eib′′

= erer+1⋯ei−1eiei+1eiei+2⋯en ⊗ b′′

= erer+1⋯ei−1eiei+2⋯en ⊗ b′′

= ei+2ei+3⋯en ⊗ erer+1⋯eib′′

= ei+2ei+3⋯en ⊗ erer+1⋯ei−1e′b.

Now set b′ ∶= erer+1⋯ei−1e′b. Then b′ and e′b do not differ on the vertices larger than i + 2

and hence b′ has no simple links at j ≥ i + 2. Setting s = i + 2 and e = ei+2ei+3⋯en closes theargument.

Let In,p be the set consisting of the elements 1⊗ b for b an (n, p)-link state and erer+1⋯en⊗ bwith 1 ≤ r ≤ n and b an r-admissible (n, p)-link state. The previous lemma guarantees that In,pis a spanning set of Vn,p↑.

Example. The set I4,1 consists of

I4,1 = 1⊗ ,1⊗ ,1⊗ , e4 ⊗ , e4 ⊗ , e4 ⊗ , e3e4 ⊗ , e3e4 ⊗ , e2e3e4 ⊗ .


In the next proposition we will use this set to prove an isomorphism between restricted andinduced modules. Besides it will prove that In,p is not only an spanning set, but even a basisof Vn,p↑.3.9 Proposition. When (n, p) ≠ (2,1) or β ≠ 0 the TLn-modules Vn−1,p↑ and Vn+1,p+1↓ are isomor-phic.

Proof. Define a map Vn−1,p → Vn+1,p+1 ∶ b ↦ b∗ by adding two vertices below b and connectingthem with a simple link (see figure 3.3 from b′ to b′∗). This induces a map

Φ ∶ Vn−1,p↑ → Vn+1,p+1↓ ∶ e⊗ b↦ eb∗,

for e ∈ TLn(β) and b ∈ Vn−1,p. We will show that Φ is an isomorphism.First we prove it is well-defined. Using the definition of Vn−1,p as a tensor product over

the complex numbers (see definition 3.7), it suffices to show that Φ(ee′ ⊗ b) = Φ(e ⊗ e′b) fore ∈ TLn, e

′ ∈ TLn−1, b ∈ Vn−1,p. But this is immediate, because both equal ee′b∗ ∈ Vn+1,p+1.Moreover, it is a homomorphism since

Φ(e(e′ ⊗ b)) = Φ(ee′ ⊗ b) = ee′b∗ = eΦ(e′ ⊗ b)

for e, e′ ∈ TLn and b ∈ Vn−1,p.Since the (n+1, p+1)-link states from a basis for Vn+1,p+1↓ as remarked below definition 3.5,

showing that each of these link states has a pre-image in In−1,p under Φ will prove surjectivity.Let b be a (n+1, p+1)-link state, then it must have at least one simple link, since it has at leastone link and no defects can occur within links. Delete the simple link at the highest numberedvertex, say, r to obtain a (n − 1, p)-link state b′. Then we have Φ(erer+1⋯en ⊗ b′) = b, see figure3.3. Since there are no simple links in the lower box, b′ is r-admissible and erer+1⋯en ⊗ b′ ∈In−1,p.

b =

n+1

r

1

b′ =

n−1

r

1

b′∗ =

n+1

r

1

Φ(er⋯en ⊗ b′)= er⋯enb′∗ =

n+1

r

1

Figure 3.3: Construction of the pre-image of b under Φ.

Finally we show injectivity of Φ. Since In−1,p is a spanning set of Vn−1,p↑ it suffices to showthat Φ maps In−1,p injectively into Vn+1,p+1↓. Let a = erer+1⋯en−1 ⊗ b and a′ = eses+1⋯en−1 ⊗ b′be elements of In−1,p (thus b is r-admissible and b′ is s-admissible) and suppose Φ(a) = Φ(a′).Then we can depict them as two blocks separated by a simple link, which is the simple linkat the highest numbered vertex, r and s respectively. Since they are equal, these simple linksmust occur at the same height, hence r = s. Now, in order to be equal, the lower boxes mustbe the same, and so must the upper boxes, proving that a = a′.

3.10 Corollary. The set In,p is a basis for Vn,p↑.3.11 Corollary. The dimension of Vn,p↑ is

dimVn,p↑ = dimVn+2,p+1↓ = dn+2,p+1,

except when (n, p) = (2,1) and β = 0, in which case dimVn,p↑ = 3.


We conclude the section with a second exact sequence.

3.12 Corollary. When (n, p) ≠ (2,1) or β ≠ 0, the sequence

0 Vn,p+1 Vn−1,p↑ Vn,p 0. (3.4)

is exact.

Proof. Recall from proposition 3.6 the exact sequence 0→ Vn,p+1 → Vn+1,p+1↓→ Vn,p → 0. Usingthe isomorphism Φ ∶ Vn+1,p+1↓ ≅ Vn−1,p↑ we find that the sequence in (4.25) is exact.

3.2 Spin chain representations

Roughly speaking, a spin chain is a number of molecules lined up. Each molecule is in a statethat is described by a value in C2. States of neighbouring molecules are influenced by eachother. The Temperley-Lieb algebra acts on the set of states. How one may view the spin chainas a theoretical physical model is treated more extensively in section 5.1.

The spin representation differs from the previous representations in that it is defined on thevector space (C2)⊗n = C2 ⊗⋯⊗C2 (n copies) rather than a formal vector space over a basis oflink state-diagrams.

The treatment is based on lecture notes by stokman [27] and articles by Morin-Duchesne etal. [22], [23].

3.2.1. The spin representation

Consider the 4-dimensional space C2 ⊗C2. The set v+, v− = (1,0), (0,1), is a basis for C2

and(v+ ⊗ v+), (v+ ⊗ v−), (v− ⊗ v+), (v− ⊗ v−)

forms a basis for C2 ⊗C2. We can represent a linear operator B ∶ C2 ⊗C2 by a (4 × 4)-matrixwith respect to the given basis. For the space (C2)⊗n = C2 ⊗ ⋯ ⊗ C2, denote by Bi,j (with1 ≤ i ≠ j ≤ n) the linear operator which acts as B on the i-th and j-th component of the n-foldtensor product and as the identity on the other components. Then Bi,j ∈ End ((C2)⊗n).

Recall β = q+q−1. We are now ready to define the spin representation on the Temperley-Liebalgebra.

3.13 Lemma. The map ζ ∶ TLn → End ((C2)⊗n) given by

ζ(ei) ∶=⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

∈ End ((C2)⊗n)

satisfies the defining relations of TLn from definition 2.6, hence it defines a representation of TLn. TheC-vector space (C2)⊗n equipped with the map ζ is called the spin representation or spin chain-module.

Proof. We check the defining relations one by one.

3.2. Spin chain representations 25

(i) We have

ζ(ei)2 =⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠

2

i,i+1

=⎛⎜⎜⎜⎝

0 0 0 00 q2 + 1 q + q−1 00 q + q−1 1 + q−2 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

= (q + q−1) ⋅ ζ(ei).

(ii) If i ≠ j ± 1 then ζ(ei) and ζ(ej) act on different components of the tensor product, hencethey commute and ζ(ei)ζ(ej) = ζ(ej)ζ(ei).

(iii) The identity ζ(ei)ζ(ei±1)ζ(ei) = ζ(ei) can be verified using a straightforward but tediouscomputation with (8 × 8)-matrices, which will be omitted.

This completes the proof.

3.14 Remark. Denote by σ− and σ+ the actions on C2 given by the matrices

σ− = (0 01 0

) , σ+ = (0 10 0

) .

Besides, consider the Pauli spin operators

σx = (0 11 0

) , σy = (0 −ii 0

) and σz = (1 00 −1

) . (3.5)

Define the map σαi ∈ End ((C2)⊗n) for 1 ≤ i ≤ n by

σ±i = id⊗⋯⊗ id´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶i−1 times

⊗σα ⊗ id⊗⋯⊗ id´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶n−i times

,

where α ∈ +,−, x, y, z. Clearly σαi and σα′

j (with α,α′ ∈ +,−, x, y, z) commute when i ≠ j.We can write ζ(ei) as

ζ(ei) = σ−i σ+i+1 + σ+i σ−i+1 − (q + q−1)σ+i σ−i σ+i+1σ−i+1 + qσ+i σ−i + q−1σ+i+1σ

−i+1. (3.6)

This can be easily verified by evaluating the basis elements of C2 (it suffices to look solely atthe i-th and (i + 1)-th component of (C2)⊗n since ζ(ei) only acts on those components).

Moreover, we can write

ζ(ei) =1

2(σxi σxi+1 + σyi σ

yi+1) −

1

4(q + q−1)(σzi σzi+1 − id) + 1

4(q − q−1)(σzi − σzi+1) (3.7)

which can be seen similarly.

Define Sz = ∑ni=1 σzi ∈ End ((C2)⊗n). Then for εi ∈ +,− we have

Sz(vε1 ⊗⋯⊗ vεn) = (n

∑i=1

εi)(vε1 ⊗⋯⊗ vεn)

Set En,p = (C2)⊗n∣Sz=p

. Then En,p is the eigenspace of (C2)⊗n corresponding to the eigenvaluep. We have

(C2)⊗n =n

⊕p=−n

n−p∈2Z

En,p.

It is clear that dimEn,p = ( n(n+p)/2

) = ( n(n−p)/2

). The following observation is easy. Let p ≤ 0, then

dimEn,p =(n+p)/2

∑i=0

dimVn,i. (3.8)


3.2.2. First link-spin intertwiner

Writing β = q+q−1, for generic q and n even we can find a homomorphism of TLn(β)-modulesVn,n/2 → (C2)⊗n. This is based on an analogous construction for the affine Temperley-Liebalgebra (treated in subsection 6.2.1) that can be found in lecture notes by Stokman [27]. Beforewe can define the homomorphism, we need some definitions.

3.15 Definitions. Recall that Bn,p is a canonical basis for Vn,p and can be viewed as the setof (equivalence classes) of (n, p)-link states. We can orientate a link by choosing its startingpoint and endpoint and we can orientate an element of Bn,p by choosing an orientation foreach link. Let Bn,p be the set of all oriented link states on n points. Let Forg ∶ Bn,p → Bn,p bethe function that forgets the orientation. For w ∈ Bn,p and j ∈ 1, . . . , n define

rj(w) ∶=⎧⎪⎪⎪⎨⎪⎪⎪⎩

+ if the link at j is outgoing− if the link at j is incoming+ if w has a defect at j

and

or(w) ∶= # links from i to j with 1 ≤ j < i ≤ n − # links from i to j with 1 ≤ i < j ≤ n .

In this subsection we specialise to p = n/2. A link state w ∈ Bn,n/2 has no defects. Thefollowing proposition gives an intertwiner between the link state-module and the spin chainrepresentation.

3.16 Proposition. Define Ψ ∶ Vn,n/2 → (C2)⊗n, linear, via

Ψ(w) ∶= ∑w∈Forg−1(w)

q−or(w)/2 ⋅ vr1(w) ⊗⋯⊗ vrn(w).

Then Ψ is an intertwined of TL-modules and injective for generic q.

Proof. Let w ∈ Bn,n/2, i ∈ 1, . . . , n. We need to check that

Ψ(µ(ei)w) = ζ(ei)Ψ(w).

We subdivide this into the following cases:

(1) Ψ(µ(ei)w) = ζ(ei)Ψ(w) for 1 ≤ i < n and i is connected to i + 1 in L.

(2) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i+ 1 is connected to k and 1 ≤ j < i <i + 1 < k ≤ n.

(3) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i + 1 is connected to k with 1 ≤ k <j < i < i + 1 ≤ n.

(4) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i + 1 is connected to k with 1 ≤ i <i + 1 < k < j ≤ n.

Let us draw only the vertices i, i+1, j and k. Then pictorically, these cases look like in figure3.4.

Let us proceed.


i

i + 1

k

i + 1i

j

i + 1i

j

k

j

k

i + 1i

Figure 3.4: Case distinction.

Case 1. If 1 ≤ i < n and i↔ i + 1 in w, then Ψ(µ(ei)w) = Ψ((q + q−1)w) = (q + q−1)Ψ(w). Forw ∈ Forg−1(w) denote by w′ the tensor product w with the i-th and (i+1)-th term interchanged.Note that or(w) = or(w′) + 2. Then

ζ(ei)Ψ(w) = ∑w∈Forg−1(w)

ri(w)=+

(q−or(w)/2 ⋅⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

(⋯⊗ v+®i-th term

⊗v− ⊗⋯)

+ q−or(w′)/2 ⋅

⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

(⋯⊗ v− ⊗ v+ ⊗⋯))

= ∑w∈Forg−1(w)

ri(w)=+

(q−or(w)/2 ⋅ q ⋅ (⋯⊗ v+ ⊗ v− ⊗⋯) + q−or(w)/2(⋯⊗ v− ⊗ v+ ⊗⋯)

+ q−or(w′)/2 ⋅ q−1 ⋅ (⋯⊗ v− ⊗ v+ ⊗⋯) + q−or(w

′)/2 ⋅ (⋯⊗ v+ ⊗ v− ⊗⋯))


ri(w)=+

(q−or(w)/2 ⋅ q ⋅ (⋯⊗ v+ ⊗ v− ⊗⋯) + q−or(w′)/2 ⋅ q−1 ⋅ (⋯⊗ v− ⊗ v+ ⊗⋯)

+ q−or(w′)/2 ⋅ q−1 ⋅ (⋯⊗ v− ⊗ v+ ⊗⋯) + q−or(w)/2 ⋅ q ⋅ (⋯⊗ v+ ⊗ v− ⊗⋯))


ri(w)=+

(q−or(w)/2(q + q−1) ⋅ (⋯⊗ v+ ⊗ v− ⊗⋯)

+ q−or(w)/2(q + q−1) ⋅ (⋯⊗ v− ⊗ v+ ⊗⋯))


ri(w)=+

q−or(w)/2(q + q−1) ⋅ vr1(w) ⊗⋯⊗ vrn(w)

+ ∑w∈Forg−1(w)

ri(w)=−

q−or(w)/2(q + q−1) ⋅ vr1(w) ⊗⋯⊗ vrn(w)

= (q + q−1)Ψ(w)= ζ(ei)Ψ(w).

Thus we find that Ψ(µ(ei)w) = ζ(ei)Ψ(w).

Case 2. Assume i is connected to j and i + 1 is connected to k and 1 ≤ j < i < i + 1 < k ≤ n.Then µ(ei)w = w′ is the matching with i↔ i+ 1 and j ↔ k and equal to w on all other vertices


(cf figure 3.5).

n

k

i + 1i

j

1

n

k

i + 1i

j

1

Figure 3.5: Ad case 2.

Let us write only the j-th, i-th, (i+ 1)-th and k-th term (in that order) of the tensor productin the image of Ψ. The the sum over all w ∈ Forg−1(w) can be splitted in four sums by orderingthe signs of rj(w) and ri+1(w). We get

Ψ(w′) = ∑w′∈Forg−1(w′)

rj(w′)=+,ri+1(w′)=+

(q−or(w′)/2 ⋅ (v+ ⊗ v− ⊗ v+ ⊗ v−))

+ ∑w′∈Forg−1(w′)

rj(w′)=+,ri+1(w′)=−

(q−or(w′)/2 ⋅ (v+ ⊗ v+ ⊗ v− ⊗ v−))

+ ∑w′∈Forg−1(w′)

rj(w′)=−,ri+1(w′)=+

(q−or(w′)/2 ⋅ (v− ⊗ v− ⊗ v+ ⊗ v+))

+ ∑w′∈Forg−1(w′)

rj(w′)=−,ri+1(w′)=−

(q−or(w′)/2 ⋅ (v− ⊗ v+ ⊗ v− ⊗ v+))

= ∑w′∈Forg−1(w′)

rj(w′)=+,ri+1(w′)=+

(q−or(w′)/2 ⋅ (v+ ⊗ v− ⊗ v+ ⊗ v−)

+ qor(w′)/2 ⋅ q ⋅ (v+ ⊗ v+ ⊗ v− ⊗ v−)

+ qor(w′)/2 ⋅ q−1 ⋅ (v− ⊗ v− ⊗ v+ ⊗ v+)

+ qor(w′)/2 ⋅ (v− ⊗ v+ ⊗ v− ⊗ v+)).

Similarly we write

Ψ(w) = ∑w∈Forg−1(w)

rj(w)=+,ri+1(w)=−

(q−or(w)/2 ⋅ (v+ ⊗ v− ⊗ v− ⊗ v+)

+ q−or(w)/2 ⋅ q ⋅ (v+ ⊗ v− ⊗ v+ ⊗ v−)+ q−or(w)/2 ⋅ q−1 ⋅ (v− ⊗ v+ ⊗ v− ⊗ v+)

+ q−or(w)/2 ⋅ (v− ⊗ v+ ⊗ v+ ⊗ v−))


Now we compute

ζ(ei)Ψ(w) = ∑w∈Forg−1(w)

rj(w)=+,ri+1(w)=−

(q−or(w)/2 ⋅⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

(v+ ⊗ v− ⊗ v− ⊗ v+)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0

+ q−or(w)/2 ⋅ q ⋅⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

(v+ ⊗ v− ⊗ v+ ⊗ v−)

+ q−or(w)/2 ⋅ q−1 ⋅⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

(v− ⊗ v+ ⊗ v− ⊗ v+)

+ q−or(w)/2 ⋅⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

(v− ⊗ v+ ⊗ v+ ⊗ v−))

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0


rj(w)=+,ri+1(w)=−

(q−or(w)/2 ⋅ (v+ ⊗ v− ⊗ v+ ⊗ v−)

+ q−or(w)/2 ⋅ q ⋅ (v+ ⊗ v+ ⊗ v− ⊗ v−)+ q−or(w)/2 ⋅ (v− ⊗ v+ ⊗ v− ⊗ v+)

+ q−or(w)/2 ⋅ q−1 ⋅ (v− ⊗ v− ⊗ v+ ⊗ v+))

Note that q−or(w′)/2 = q−or(w)/2 when w′ ∈ Forg−1(w′) with rj(w′) = + and ri+1(w′) = + and

w ∈ Forg−1(w) with rj(w) = + and ri+1(w) = −. This yields

Ψ(µ(ei)w) = Ψ(w′) = ζ(ei)Ψ(w).

Case 3 and 4. The last two cases are very similar to case 2.

Injectivity. We have or(w) ∈ −n,1 − n, . . . , n − 1, n and hence or(w)/2 ∈ −n2, . . . , n

2 for all

(n, p)-link states w. Therefore qn/2Ψ(w) is polynomial in q1/2 and

qn/2Ψ ∶ Vn,n/2 → (C2)⊗n

is linear over C.Lift qn/2Ψ to a polynomial

Ψ ∶ C[s][Bn,n/2]→ C[s]⊗ (C2)⊗n

defined byΨ(w) = ∑

w∈Forg−1(w)

sn−or(w) ⊗ vr1(w) ⊗⋯⊗ vrn(w).


Then Ψ is linear of C[v] and the evaluation v = q1/2 yields the polynomial qn/2Ψ.We can view Ψ as a matrix whose entries are polynomial in v. Suppose Ψ is injective, then

Ψ has a maximal square submatrix with nonzero determinant. The determinant is a nonzeropolynomial in v, hence it is nonzero for all v except when v is a root of the determinant. Thisimplies that evaluation of Ψ in v = q1/2 yields a matrix with a maximal square submatrix ofnonzero determinant for generic q, (the only exceptions occurring when q1/2 is a root of thedeterminant). Thus qn/2Ψ is injective for generic q. Consequently, multiplying by q−n/2 showsthat Ψ is injective for generic q.

Now in order to prove injectivity of Ψ for generic q, it suffices to show Ψ is injective. Letw ∈ Bn,n/2, then Ψ(w)∣

v=0gives the basis vector of (C2)⊗n corresponding to the orientation of

w with or(w) = n. Let us call this term vwmax . Clearly, when w ≠ w′ are (n, p)-link states, vwmax

and vw′max

are different basis elements of (C2)⊗n. Suppose v = ∑w∈Bn,n/2 cww ∈ C[s][Bn,n/2]and suppose Ψ(v) = 0. Then

Ψ(∑w

cww)∣v=0

=∑w

(cwΨ(w)∣v=0

) =∑w

cwvwmax = 0,

and since all vwmaxare different basis elements of (C2)⊗n the cw must be zero for all w. We

conclude that v = 0 so that Ψ is injective for generic q.

Proposition 3.16 gives an injective homomorphism Ψ ∶ Vn,n/2 → (C2)⊗n which is easilyseen to land in En,0. One could hope that this restriction turns Ψ into an isomorphism.Unfortunately, the inequality dimCVn,n/2 = ( n

n/2) − ( n

n/2−1) < ( n

n/2) = dimC(C2)⊗n shows that

this is not the case. Its analog in the affine case, however, turns out to be an isomorphism (seesubsection 6.2.1).

3.2.3. Second link-spin intertwiner

This subsection is devoted to giving a second link-spin intertwiner, which individually mapseach of the Vn,p’s into (C2)⊗n. Its analogue for the affine Temperley-Lieb algebra was firstgiven by Morin-Duchesne and Saint-Aubin in [23]. This intertwining operator turns out to bea generalisation of Ψ and can be written in a similar fashion using a sum over oriented linkstate-diagrams. The sum-notation is not given in the article.

3.17 Proposition. Suppose β = q + q−1 and u ∈ C× is such that u2 = q. For w ∈ Bn,p, set ψ(w) ∶=(j1, j′1), . . . , (jp, j′p), where 1 ≤ jk < n denotes the beginning of a link and j′k with jk < j′k ≤ n thecorresponding endpoint. Write (v⊗n+ ) = (v+ ⊗⋯⊗ v+). Define the map Ωn,p ∶ Vn,p → (C2)⊗n by

Ωn,p(w) = ∏(j,j′)∈ψ(w)

(uσ−j′ + u−1σ−j )(v⊗n+ )

for w ∈ Bn,p and linearly extending it to Vn,p. Then Ω intertwines µ and ζ, that is,

Ωn,p(µn,p(x)w) = ζ(x)Ωn,p(w)

for all x ∈ TLn(β) and w ∈ Vn,p.

Before proving the proposition we make a remark on the notation of the intertwiner anddraw a corollary. This proposition may be formulated differently to resemble the previousintertwiner more. To this end, recall definitions 3.15.


3.18 Lemma. For w ∈ Bn,p we have

Ωn,p(w) = ∑w∈Forg−1(w)

q−or(w)/2vr1(w) ⊗⋯⊗ vrn(w).

Proof. Let Ωn,p(w) = ∑w∈Forg−1(w) q−or(w)/2vr1(w) ⊗⋯⊗ vrn(w). It is clear that both Ωn,p(w) and

Ωn,p(w) sum over the same pure tensors. Let v = (vε1 ⊗ ⋯ ⊗ vεn) be a term in the sum ofΩn,p(w) and suppose k (with 0 ≤ k ≤ p) of the components are the left vertex of a link. Thenor(v) = k − (p − k) = 2k, so q−or(v)/2 = qp/2−k = up−2k. On the other hand, in Ωn,p(w) we get afactor u−1 for each of the v−’s at the left vertex of a link, and a factor u for the other ones. Thiscomes to a factor (u−1)kup−k = up−2k. Thus the notations indeed coincide.

With this lemma the following corollary is immediate.

3.19 Corollary. The TLn(β)-module Ω is a generalisation of the TLn(β)-module Ψ. That is, theaction of Ωn,n/2 on En,0 is the same as that of Ψ.

Now let us give a proof of proposition 3.17. For clarity, we will drop the subscript n, p inthe proof of the proposition.

Proof of proposition 3.17. The case x = id is trivial. It suffices to check x = ei for 1 ≤ i ≤ n − 1acting on a link state w ∈ Bn,p. The general result then follows by linearity.

If w has a link from i to i+1, (has a simple link at i) we have Ω(µ(ei)w) = βΩ(w). By a directcomputation using remark 3.14 to write ζ(ei) we find this to equal ζ(ei)Ω(w). An elaborateexample of this computation is done in step 3 below.

If b has a defect at i and at i+1 then all terms in Ω(w) will have a v+ at component i and i+1.The action of ζ(ei) maps this to 0, hence ζ(ei)Ω(w) = 0. This is equal to Ω(µ(x)w) = Ω(0) = 0.

Apart from these two, there are five more cases for w, depicted in figure 3.6. Only thevertices i, i + 1, j and k are shown, where i is connected to j and i + 1 to k (if they are not adefect). One can easily see that all other vertices do not contribute to the computation.

w =i + 1

i

j

k

i + 1i

k

i + 1i

j

i + 1i

j

k

j

k

i + 1i

Figure 3.6: Possible w in Bn,p. Case 3 through 7.

Note that σ+i σ−i (v⊗n+ ) = (v⊗n+ ), σ+i (v⊗n+ ) = 0 and σ−i σ

−i (v⊗n+ ) = 0. Denote by Yi(w) the product

∏(uσ−d′ + u−1σ−d) over all (d, d′) ∈ ψ(b) that do not involve vertex i and i + 1. (This is well-defined because all σ−i commute with each other.) Now let us consider the cases. Since allcomputations are a matter of writing down definitions, we will elaborate case 3 and go troughthe other cases rather quickly.


Case 3. Compute

ζ(ei)Ω(w)= ζ(ei)(uσ−i + u−1σ−j )Yi(w)(v⊗n+ )= (σ−i σ+i+1 + σ+i σ−i+1 − (q + q−1)σ+i σ−i σ+i+1σ

−i+1 + qσ+i σ−i + q−1σ+i+1σ

−i+1)

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=ζ(ei)

⋅ (uσ−j + u−1σ−i )Yi(w)(v⊗n+ )

= (σ−i σ+i+1u−1σ−i

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0

+σ+i σ−i+1uσ−i

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=uσ−i+1

−(q + q−1)σ+i σ−i σ+i+1σ−i+1uσ

−i

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0

+ q σ+i σ−i uσ−i´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

=0

+q−1 σ+i+1σ−i+1uσ

−i

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=uσ−i

)Yi(w)(v⊗n+ )

+ (σ−i σ+i+1 + σ+i σ−i+1´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

=0

−(q + q−1)σ+i σ−i σ+i+1σ−i+1

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=id

+q σ+i σ−i²=id

+q−1 σ+i+1σ−i+1

´¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¶=id

)u−1σ−j Yi(w)(v⊗n+ )

= (uσ−i+1 + u−1σ−i )Yi(w)(v⊗n+ )= Ω(µ(ei)w).

Case 4. Similar to case 3.

Case 5. Using the same tricks as in case 3, compute

ζ(ei)Ω(w) = ζ(ei)(uσ−i + u−1σ−j )(uσ−k + u−1σ−i+1)Yi(w)(v⊗n+ )= ζ(ei)(qσ−i σ−k + σ−i σ−i+1 + σ−j σ−k + q−1σ−j σ

−i+1)Yi(w)(v⊗n+ )

= (qσ−i+1σ−k + σ−i σ−k + q−1σ−i σ

−j + σ−j σ−i+1)Yi(w)(v⊗n+ )

= (uσ−i+1 + u−1σ−i )(uσ−k + u−1σ−j )Yi(w)(v⊗n+ )= Ω(µ(ei)w).

Case 6. We have

ζ(ei)Ω(w) = ζ(ei)(uσ−i + u−1σ−j )(uσ−i+1 + u−1σ−k)Yi(w)(v⊗n+ )= ζ(ei)(qσ−i σ−i+1 + σ−i σ−k + σ−j σ−i+1 + q−1σ−j σ

−k)Yi(w)(v⊗n+ )

= (σ−i+1σ−k + q−1σ−i σ

−k + σ−i σ−j + qσ−j σ−i+1)Yi(w)(v⊗n+ )

= (uσ−j + u−1σ−k)(uσ−i+1 + u−1σ−i )Yi(w)= Ω(µ(ei)w).

Case 7. Similar to case 6.

This finishes the proof. We conclude that Ω is an intertwining operator.

3.3 The dimer representations

In this section we represent TLn(β) on (C2)⊗(n−1). In our approach we follow an article byMorin-Duchesne et al. [22] who defined the dimer representation of TLn(0).

3.3. The dimer representations 33

Recall the definition of σ±i from 3.14.

σ− = (0 01 0

) , σ+ = (0 10 0

) .

Define the dimer representation as follows. Let σ±0 ≡ σ±n ≡ 0.

3.20 Lemma. Let β = 0. The map τ ∶ TLn → End ((C2)⊗(n−1)) generated by

τ(ei) = σ−i−1σ+i + σ+i σ−i+1.

uniquely defines a TLn(0)-representation called the dimer representation.

Proof. By remark 3.2 it suffices to check that τ satisfies the TL-relations from (2.2). This iseasily done. First, we use (σ±i )2 = 0 to find

τ(ei)2 = (σ−i−1σ+i + σ+i σ−i+1)2

= (σ−i−1)2(σ+i )2 + (σ+i )2(σ−i+1)2 + 2 ⋅ (σ−i−1σ+i σ

+i σ

−i+1)

= 0.

Note that σ±i σ∓i σ

±i = σ±i . Using this and the fact that we may permute σ’s if their indices

differ, we explicitly compute

τ(ei)τ(ei+1)τ(ei) = (σ−i−1σ+i + σ+i σ−i+1)(σ−i σ+i+1 + σ+i+1σ

−i+2)(σ−i−1σ

+i + σ+i σ−i+1)

= σ−i−1σ+i σ

−i σ

+i+1σ

−i−1σ

+i + σ−i−1σ

+i σ

−i σ

+i+1σ

+i σ

−i+1

+ σ−i−1σ+i σ

+i+1σ

−i+2σ

−i−1σ

+i + σ−i−1σ

+i σ

+i+1σ

−i+2σ

+i σ

−i+1

+ σ+i σ−i+1σ−i σ

+i+1σ

−i−1σ

+i + σ+i σ−i+1σ

−i σ

+i+1σ

+i σ

−i+1

+ σ+i σ−i+1σ+i+1σ

−i+2σ

−i−1σ


+i+1σ

−i+2σ

+i σ

−i+1

= 0 + σ−i−1σ+i σ

+i+1σ

−i+1 + 0 + 0

+ σ−i−1σ+i σ

−i+1σ

+i+1 + σ+i σ−i+1 + 0 + 0

= σ−i−1σ+i (σ+i+1σ

−i+1 + σ−i+1σ

+i+1

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=id

) + σ+i σ−i+1

= σ−i−1σ+i + σ+i σ−i+1

= τ(ei).

In a similar fashion we obtain τ(ei)τ(ei−1)τ(ei) = τ(ei).Finally, let ∣i − j∣ > 1 and compute

τ(ei)τ(ej) = (σ−i−1σ+i + σ+i σ−i+1)(σ−j−1σ

+j + σ+j σ−j+1)

= σ−i−1σ+i σ

−j−1σ

+j + σ+i σ−i+1σ

−j−1σ

+j + σ−i−1σ

+i σ

+j σ

−j+1 + σ+i σ−i+1σ

+j σ

−j+1

= σ−j−1σ+j σ

−i−1σ


−j−1σ

+j + σ−i−1σ

+i σ

+j σ

−j+1 + σ+j σ−j+1σ

+i σ

−i+1

= (σ−j−1σ+j + σ+j σ−j+1)(σ−i−1σ

+i + σ+i σ−i+1)

= τ(ej)τ(ei).

Note that we could not safely permute σi±1 and σj∓1 since their indices might be equal, butwe did not need to.


Example. Consider (v+ ⊗ v− ⊗ v− ⊗ v−) ∈ (C2)⊗4 and let e2 ∈ TL5(0) act on it. Then we get

τ(e2)(v+ ⊗ v− ⊗ v− ⊗ v−) = (σ−1σ+2 + σ+2σ−3 )(v+ ⊗ v− ⊗ v− ⊗ v−)= (σ−v+ ⊗ σ+v− ⊗ v− ⊗ v−) + (v+ ⊗ σ+v− ⊗ σ−v− ⊗ v−)= (v− ⊗ v+ ⊗ v− ⊗ v−) + 0.

Restricting τ to a representation on En−1,p still yields a TLn(0)-module, denoted by τp. Wewill construct an intertwiner Γn,p between the TLn(0)-module Wn,p from lemma 3.3 and thedimer representation τn−1−p on (C2)⊗(n−1).

Let w ∈ Bn,p ∪ Bn,p−1 be a link state and set ψ(w) ∶= (j1, j′1), . . . , (jm, j′m), where 1 ≤ jk < ndenotes the beginning of a link and j′k with jk < j′k < n + jk the corresponding endpoint. Setσ0 ≡ σn ≡ 0 and define the map Γp by

Γp(w) ∶= ∏(j,j′)∈ψ(w)

(σ−j−1 + σ−j′)(v⊗(n−1)+ )

when w ∈ Bn,p andΓp(w) ∶= σ`(w)−1 ∏

(j,j′)∈ψ(w)

(σ−j−1 + σ−j′)(v⊗(n−1)+ )

when w ∈ Bn,p−1, where `(b) denotes the number of the defect on the highest numberedvertex. Linearly extending this map to all of Wn,p yields the map

Γp ∶Wn,p → En−1,n−1−2p.

The following proposition proves that this is an intertwiner between the link state moduleW∗,∗ and the dimer representation on (C2)⊗(n−1).

3.21 Proposition. The map Γp intertwines the representations θp and τn−1−2p, that is,

Γp(θp(x)w) = τn−1−2p(x)Γp(w)

for all x ∈ TLn(0) and w ∈Wn,p.

Proof. It suffices to show the intertwining property for x = 1 and x = ei, acting on link statesin Bn,p ∪ Bn,p−1. The general result then follows by linearity of Γp, θp and τp.

For x = 1 the identity is trivial. When x = ei is acting on a link state in Bn,p, there are sevencases for w ∈ Bn,p, similar to those in the proof of proposition 3.17. These are drawn in figure3.8, where only the vertices connected to i and i + 1 are depicted.

w =i + 1

i

i + 1i

i + 1i

j

k

i + 1i

k

i + 1i

j

i + 1i

j

k

j

k

i + 1i

Figure 3.7: Cases 1 to 7.

Each case translates into an algebraic identiy. Let us go through them one by one. As in theproof of proposition 3.17, let Yi(w) denote the product ∏(σ−d−1 + σ−d′) over all (d, d′) ∈ ψ(w)that do not involve vertex i and i + 1.

3.3. The dimer representations 35

Case 1. When w has defects in both i and i + 1, we get

Γp(θp(ei)w) = Γp(0) = 0 = (σ−i−1σ+i + σ+i σ−i+1)Yi(w)(v⊗(n−1)

+ ) = τn−1−2p(ei)Γp(w),

where the third equality holds since Yi(w)(v⊗(n−1)+ ) has a v+ on the i-th position of each

summand and σ+(v+) = 0.

Case 2. If w has a link from i to i + 1, then θ(ei)w = 0 and the same argument as in case 1

holds.

Case 3. When j is connected to i, j < i and w has a defect at i + 1, we find

Γp(θp(ei)w) = Yi(w)(σ−i−1 + σ−i+1)(v⊗(n−1)+ )

= Yi(w)(σ−j−1σ−i−1σ

+i

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0

+σ−j−1σ+i σ

−i+1

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0

+σ−i−1 + σ+i+1)(v⊗(n−1)+ )

= Yi(w)(σ−i−1σ+i + σ+i σ−i+1)(σ−j−1 + σ−i )(v

⊗(n−1)+ )

= (σ−i−1σ+i + σ+i σ−i+1)Yi(w)(σ−j−1 + σ−i )(v

⊗(n−1)+ )

= τn−1−2p(ei)Γp(w).

In the equation, in the second equality the indicated terms are zero because of the σ+i and inthe fourth equation Y (w) and (σ−i−1σ

+i + σ+i σ−i+1) since σ−d and σ−d′ commute for all d, d′ and σ−i

does not occur in Y (w). It follows that the identity holds.

Case 4. When w has a defect in i and a link from i + 1 to k, we get the following algebraicidentity, which can be proven in a similar way as case 3,

Γp(θp(ei)w) = Yi(w)(σ−i−1 + σ−i+1)(v⊗(n−1)+ ) = (σ−i−1σ

+i + σ+i σ−i+1)Yi(w)(σ−i + σ−k)(v

⊗(n−1)+ ).

Case 5. If j is connected to i and i+ 1 to k we find a similar computation. This time we startfrom the right hand side of the identity,

τn−1−2p(ei)Γp(w) = (σ−i−1σ+i + σ+i σ−i+1)Yi(w)(σ−j−1 + σ−i )(σ−i + σ−k)(v

⊗(n−1)+ )

= Yi(w)(σ−i−1σ+i + σ+i σ−i+1)(σ−j−1σ

−i + σ−j−1σ

−k + σ−i σ−k)(v

⊗(n−1)+ )

= Yi(w)(σ−j−1σ−i−1 + σ−j−1σ

−i+1 + σ−i−1σ

−k + σ−i+1σ

−k)(v

⊗(n−1)+ )

= Yi(w)(σ−j−1 + σ−k)(σ−i−1 + σ−i+1)(v⊗(n−1)+ )

= Γp(θp(ei)w).

Case 6 and 7. The proof of these cases is very similar to the proofs above.

In order to extend to proof to all of Bn,p ∪ Bn,p−1, hence by linearity to all of Wn,p, we needto consider the cases involving a defect on the highest vertex. Let us draw those defects witha on the right side. The following w need to be considered.These cases are dealt with similarly.

Define the operator J on (C2)⊗(n−1) by J = ∑n−2i=1 (−1)i−1σ−i σ

−i+1. It is easily seen that J maps

En−1,p into En−1,p−4.

3.22 Lemma. The map J intertwines the Dimer representations τp and τp−4, that is, Jτp(x) =τp−4J(x) for all x ∈ TLn(0).


w =i + 1

i

i + 1i

j

k

i + 1i


Proof. For x = 1 the statement is trivial. Let x = ei, the we find that Jτp(ei) consists of theterms (−1)j−1σ−j σ

−j+1(σ−i−1σ

+i + σ+i σ−i+1) for j = 1,2, . . . , n − 2. When j ∉ i − 1, i this is equal to

(−1)j−1(σ−i−1σ+i +σ+i σ−i+1)σ−j σ−j+1, since σ−dσ

−d′ = σ−d′σ−d for all d, d′ and σ−dσ

+d′ = σ+d′σ−d for all d ≠ d′.

Let us separately consider the cases j = i − 1 and j = i. When j = i − 1 we find

σ−i−1σ−i (σ−i−1σ

+i + σ+i σ−i+1) = σ−i−1σ

−i σ

+i σ

−i+1,

and for j = iσ−i σ

−i+1(σ−i−1σ

+i + σ+i σ−i+1) = σ−i−1σ

−i σ

+i σ

−i+1.

Since the signs in the sum alternate, these cancel.In a similar way we find that the terms for j = i − 1 and j = i in the sum of τp−4J cancel, so

thatJτp(ei) = τp−4(ei)J

and by linearity the general case follows.

Since JkΓp now defines an intertwining operator Wn,p → En−1,n−1−2p−4k, the map hp,k ∶=JkΓ(n−1−p−2k)/2 defines a maps from the composite module Wn,(n−1−p−4k)/2 into En−1,p for all0 ≤ 4k ≤ nn − 1 − p. The following lemma proves an important property of these hp,k, inparticularly proving they are nonzero.

3.23 Lemma. The map hp,k is an intertwiner and is nonzero on the submodule V(n−1−p−2k)/2 ⊂W(n−1−p−2k)/2.

Proof. The map hp,k is an intertwiner because Γp and J both are. The image of Vn,(n−p)/2−2k−1/2

is nonzere, because the element in figure 3.9 is nonzero under hp,k.

Figure 3.9: Element in V(n−p)/2−2k−1/2.

With help of the intertwiners above, subsection 5.2.2 uncovers the structure of the Dimerrepresentation. It is found that for n odd, the Dimer representation is isomorphic to

Ev ≅ Vn,2∣v∣+1 ⊕Vn,2∣v∣+5 ⊕ . . .⊕Vn,p′ .

For n even, the structure appears to be more involved.

3.4. Overview of TLn(β)-modules and their connections 37

3.4 Overview of TLn(β)-modules and their connections

Let us conclude this chapter with an overview of the TLn(β)-modules defined thus far. Recallthe link state modules Vn,p, Wn,p, Vn−1,p−1↑ and Vn+1,p↓ from section 3.1, where the two lattermodules are, under some mild conditions, isomorphic via Φ (proposition 3.9). When the pair(n, p) is non-critical, a short exact sequence splits the induced module into the direct sum oftwo standard modules.

In section 3.2 the spin chain module is defined and the representation homomorphismΩn,p ∶ Vn,p → En,p is is constructed, the homomorphism Ψ ∶ Vn,n/2 → En,n/2 being a specialcase.

The third section is about the dimer representation on (C2)⊗(n−1) when β = 0. The mapΓ is a module homomorphism Wn,p → En−1,n−1−2p, which yields a map Vn,p → En−1,n−1−2p.Note that the eigenspace En−1,p of (C2)⊗(n−1) now carries a different module structure thanthe spin chain representation. This assertion will become more apparent in chapter 5.

Figure 3.10 graphically represents the modules and intertwiners.

Wn,p Vn−1,p−1↑ Vn+1,p↓

En−1,n−1−2p Vn,p ⊕Vn,p−1

En,p ⊕ En,p−1

Γp β = 0∼

∼

Φ

∼ (n,p) non-critical

Ωn,p⊕Ωn,p−1

Figure 3.10: Module identities for generic q.

Chapter 4

The structure of TLn(β)

In the previous chapter we have seen that we can write Vn,p↑ ≅ Vn+1,p+1 ⊕Vn+1,p for generic q.Moreover the dimer representation was said to decompose in standard modules. These Vn,pappear to be the building blocks for some of the representations of TLn(β). As we will see inthis chapter, this is no coincidence.

This chapter is concerned with the question for which q the Temperley-Lieb algebra TLn(β)(where β = q + q−1) is semisimple. The first section is devoted to proving semisimplicity ofTLn for generic q. The second section is dedicated to the special case β = 0 and β = ±2 (q = ±iand q = ±1 respectively). The last section deals with the non-semisimple cases and classifiesthe principal indecomposables of TLn(β).

A large portion of the theory has been treated by Westbury in [29]. Yet, in this chapter wefollow a more recent and streamlined publication by Ridout and Saint-Aubin [26]. However,we do use a different central element in the proofs of some of the propositions.

4.1 Semisimplicity for generic q

Our aim in this section is to find all irreducible representations for TLn with β ≠ 0 and q notequal to a root of unity. This ultimately leads to the expression of TLn as a direct sum of itsirreducible representations in equation (4.16).

4.1.1. The radical of a standard module

We define a bilinear form on Vn,p, which gives rise to the definition of a radical on Vn,p.Besides, we n-define the diagram ∣x, y∣ which is constructed from two elements x, y ∈ Vn,p.

4.1 Definitions. (i) Let x and y be (n, p)-link states. Reflect x vertically to obtain a line withlinks and defects on both sides. If any of the defects of x is connected to another defectof x, then the form is equal to 0. If that does not occur, let k be the number of circles inthe figure and define

⟨x, y⟩ ∶= βk.

We define a symmetric bilinear form on Vn,p by extending this construction bilinearlyto Vn,p.

(ii) The radical of the above bilinear form on Vn,p is

Rn,p ∶= x ∈ Vn,p ∣ ⟨x, y⟩ = 0 for all y ∈ Vn,p.

(iii) Given two elements x, y ∈ Vn,p we construct a diagram ∣x, y∣ ∈ TLn by reflecting y verti-cally and connecting the defects of x, y. Note that there is a unique way to connect thedefects. Extend this construction linearly to all of Vn,p.

Before we continue our study of representations, we have a look at some examples.

39

40 Chapter 4. The structure of TLn(β)

Example. Denote the basis of V5,2 given in figure 3.1 by v1, . . . , v5. Then ⟨v1, v2⟩ = β, ⟨v4, v5⟩ = 1and

∣v2, v3∣ = , ∣v4, v1∣ = .

Denote by x† the vertical reflection of the diagram x ∈ TLn. The map x ↦ x† is an antiau-tomorphism of TLn. Define v† where v ∈ Vn,p in a similar way. The bilinear form defined indefinition 4.1 is invariant with respect to the action from TLn. We easily see that

⟨v, xw⟩ = ⟨x†v,w⟩ (4.1)

for all x ∈ TLn and v,w ∈ Vn,p.

4.2 Lemma. The notions from definition 4.1 relate to each other via

∣v,w∣u = ⟨w,u⟩v. (4.2)

Proof. By linearity it suffices to consider the case where v,w and u are (n, p)-link states. Iftwo defects of w are closed by a link in u, then ⟨w,u⟩ = 0 and two defects of v in ∣v,w∣u areclosed, so that ∣v,w∣u ∈ Vn,p+1 and ∣v,w∣u = 0 in Vn,p. If no defects are closed, ∣v,w∣u is equalto v multiplied by βk, where k is the number of circles formed by the concatination of w† andu. Of course, this is the same as ⟨w,u⟩v.

The following proposition is an important result that we will use to check irreducibility ofthe Vn,p.

4.3 Proposition. If ⟨⋅, ⋅⟩n,p is not identical to 0, then Vn,p is cyclic and indecomposable. FurthermoreVn,p/Rn,p is irreducible, i.e. Rn,p is the unique maximal proper submodule of Vn,p.

Remark. If β ≠ 0 then ⟨x,x⟩ = βp ≠ 0, so the proposition holds for all β we consider in thissection. In fact, the bilinear form is identical to 0 only if β = 0 and n = 2p, which will be dealtwith in proposition 4.16.

Proof of 4.3. Since ⟨⋅, ⋅⟩n,p ≠ 0, we can find y, z ∈ Vn,p such that ⟨y, z⟩ = 1. Then for all x ∈ Vn,pwe have ∣x, y∣z = ⟨y, z⟩x = x, so that z generates Vn,p as a TLn-module, proving that it is acyclic module since it is generated by a single element. For every z ∉ Rn,p we can find a y suchthat z is a generator. Therefore every non-zero element of Vn,p/Rn,p generates this quotient,so that Vn,p/Rn,p is irreducible.

Now assume Vn,p is decomposable, say, Vn,p = A ⊕B. Then Rn,p = (Rn,p ∩A) ⊕ (Rn,p ∩B)and we can write

Vn,p/Rn,p = (A⊕B)/((Rn,p ∩A)⊕ (Rn,p ∩B))= A/(Rn,p ∩A)⊕B/(Rn,p ∩B).

Since Vn,p/Rn,p is irreducible, either A/(Rn,p ∩ A) or B/(Rn,p ∩ B) has to be zero. AssumeA/(Rn,p ∩ A) = 0, then A ⊂ Rn,p ∩ A and A ⊂ Rn,p. But then there exists a non-zero elementz ∈ B not in Rn,p that does not generate Vn,p = A⊕B, which is a contradiction with the above.Hence Vn,p is indecomposable.

4.1. Semisimplicity for generic q 41

Set Ln,p = Vn,p/Rn,p. By the previous proposition these quotients are irreducible.

4.4 Proposition. Let N be a submodule of Vn,p and N′ of Vn,p′ and let p > p′. Suppose ⟨⋅, ⋅⟩ ≠ 0. Thenthe only module homomorphism θ ∶ Vn,p/N → Vn,p′/N′ is the zero homomorphism.

Proof. Let γ ∶ Vn,p → Vn,p/N be the canonical map. Since ⟨⋅, ⋅⟩n,p is non-zero, we can findv′,w ∈ Vn,p with ⟨v′,w⟩n,p ≠ 0. Set v ∶= v′ ⋅ 1

⟨v′,w⟩n,p, then ⟨v,w⟩n,p = 1 and for all u ∈ Vn,p

∣u, v∣θ(γ(w)) = θ(γ(∣u, v∣w)) = θ(γ(u)).

When p > p′, θ(γ(w)) has p′ links and left multiplying by ∣u, v∣ leads to extra links, thusθ(γ(u)) = ∣u, v∣θ(γ(w)) = 0. Since γ is surjective θ must also be identically zero.

Setting N = N′ = 0 shows that there exists no nonzero homomorphism Vn,p → Vn,p′ whenp ≠ p′. In particular there exists no isomorphism, so that Vn,p /≅ Vn,p′ . Setting N = Rn,pand N′ = Rn,p′ yields a similar result for Ln,p and Ln,p′ . We have now proven the followingcorollary.

4.5 Corollary. If the forms ⟨⋅, ⋅⟩n,p and ⟨⋅, ⋅⟩n,p′ on Vn,p and Vn,p′ are both non-zero, then p ≠ p′

implies Vn,p /≅ Vn,p′ and Ln,p /≅ Ln,p′ .

In a similar fashion as proposition 4.4 we can prove that every homomorphism of Vn,p intoitself is the same as a multiple of the identity.

4.6 Proposition. When ⟨⋅, ⋅⟩ ≠ 0, every module homomorphism θ ∶ Vn,p → Vn,p is a multiple of theidentity.

Proof. Pick v,w ∈ Vn,p such that ⟨v,w⟩ = 1. Then for all u ∈ Vn,p we have

θ(u) = θ(∣u, v∣w) = ∣u, v∣θ(w) = ⟨v, θ(w)⟩u,

proving the proposition.

In the next subsection, we find out when the radical Rn,p is zero, implying irreducibility ofVn,p.

4.1.2. Gram matrices

To show that Vn,p is irreducible, it suffices to show that Rn,p = 0 i.e. ⟨⋅, ⋅⟩ is non-degenerate.By linearity is suffices to check the products ⟨v,w⟩ of the (n, p)-link states, i.e. with v,w basiselements. We can construct a (dimVn,p × dimVn,p)-matrix of these forms, where on the (i, j)-entry we place the solution ⟨v,w⟩ of the i-th and j-th basis element of Vn,p. We call suchmatrices Gram matrices and denote them by Gn,p.

Example. The Gram matrix for V5,2 (cf figure 3.1) is

G5,2 =

⎛⎜⎜⎜⎜⎜⎝

β2 β 1 β ββ β2 β 1 11 β β2 β ββ 1 β β2 1β 1 β 1 β2

⎞⎟⎟⎟⎟⎟⎠

.

The radical Rn,p equals the kernel of Gn,p, so Rn,p = 0 if and only if detGn,p ≠ 0. One caneasily see that detGn,0 = 1.


4.7 Definitions. (i) Let q ∈ C/0 be given. We call a pair (n, p) critical if q2(n−2p+1) = 1.(ii) Define the number [m]q as

[m]q ∶=qm − q−mq − q−1

,

for q ≠ ±1 and [m]q =mqm−1 for q = ±1.

Note that [m]q depends on q and m continuously.

4.8 Lemma. Part (i) and (ii) of the above definition relate as follows: the pair (n, p) is critical if andonly if [n − 2p + 1]q = 0 or q = ±1.

Proof. When q = ±1 the implication to the left is trivial. For q ≠ ±1 we see [n − 2p + 1]q = 0implies q(n−2p+1) = q−(n−2p+1) so that q2(n−2p+1) = 1. On the other hand, assume q is critical,then q(n−2p+1) is its own inverse, proving the claim.

Recall the existence of the central element Jn ∈ TLn from definition 2.14. We will prove thatthe exact sequence from proposition 3.6 splits. In order to do that, we need to know that theelement Jn acts on Vn,p as the identity times gn,p = (−1)n(q(n−2p+1) + q−(n−2p+1)). The latter isproven in proposition A.1.

4.9 Proposition. If (n, p) is not critical, the exact sequence in proposition 3.6 splits, i.e. Vn,p ↓ ≅Vn−1,p ⊕Vn−1,p−1.

Proof. Given the central element Jn−1. The eigenspaces of Jn−1 are again modules of TLn−1.Since Vn−1,p and Vn−1,p−1 are indecomposable, Jn−1 has at most two eigenspaces hence at mosttwo corresponding eigenvalues gn−1,p and gn−1,p−1. If gn−1,p ≠ gn−1,p−1 both eigenspaces aresubmodules. Proposition A.1 gives gn−1,p−1 − gn−1,p = (−1)n−1(q − q−1)(qn−2p+1 − q−(n−2p+1)). Ifone of these factors is zero then (n, p) is critical, thus under our assumption the eigenvaluesare distinct and the sequence splits.

Suppose a splitting exists, then the diagram in figure 4.1 commutes. In the diagram, pridenotes the projection on the i-th coordinate.

Vn−1,p Vn,p ↓ Vn−1,p−1

Vn−1,p ⊕Vn−1,p−1

ϕ ψ

Ψpr1 pr2

# #

Figure 4.1: A splitting of the exact sequence.

If b is a basis element of Vn,p ↓ /Vn−1,p, then Ψ(b) = u + v with u ∈ Vn−1,p and v ∈ Vn−1,p−1.We know pr2 (Ψ(b)) = v = ψ(b), hence Ψ(b) = u + ψ(b). If b is a basis element of Vn−1,p andΨ(b) = u + v again, we have pr2 (Ψ(b)) = ψ(b) = 0 since b in in the image of ϕ, thereforein ker(ψ). This implies v = 0 and Ψ(b) = pr1 (Ψ(b)) = u = ϕ−1(b). This shows that we canrepresent Ψ by

Un,p = (id Vn,p0 id

) , (4.3)

where Vn,p denotes the non-trivial part of the splitting.


4.10 Lemma. If the splitting Ψ exists, define a bilinear form on Vn−1,p ⊕Vn−1,p−1 by

⟨⟨x + x′, y + y′⟩⟩ ∶= ⟨Ψ−1(x + x′),Ψ−1(y + y′)⟩n,p′ .

The above form is symmetric and invariant, so that

⟨⟨x + x′, y + y′⟩⟩ = ⟨x, y⟩n−1,p + αn,p⟨x′, y′⟩n−1,p−1′ ,

for some αn,p ∈ C.

Proof. Symmetry of this form is immediate. It is invariant in the sense of (4.1).Let V be an irreducible module over an associative algebra A and let ⟨⋅, ⋅⟩ ∶ V × V → C be

a non-zero bilinear symmetric invariant form. Then V ∗ is an A-module by (aT )(v) ∶= T (a†v)for a, v ∈ V and T ∈ V ∗. Moreover V ∗ is irreducible. The map V → V ∗ ∶ v ↦ ⟨v, ⋅⟩ is anisomorphism and is in HomA(V,V ∗). Since it is non-zero, Schur’s lemma tells us V ≃ V ∗ andthe map is fixed modulo constant multiplication.

In the situation above we have W = Vn−1,p ⊕ Vn−1,p−1, where W is the direct sum of twonon-isomorphic irreducible modules over TLn. We are given the form ⟨⟨⋅, ⋅⟩⟩ ∶ W ×W → Cwhich induces a map

ψ ∶W →W ∗.

We know W ∗ ≃ V∗n−1,p ⊕ V∗n−1,p−1, Vn−1,p ≃ V∗n−1,p and Vn−1,p−1 ≃ V∗n−1,p−1. Since the twoirreducible modules are non-isomorphic we have

Hom(Vn−1,p,V∗n−1,p−1) = 0 and Hom(Vn−1,p−1,V

∗n−1,p) = 0,

so thatHom(W,W ∗) ≃ Hom(Vn−1,p,V

∗n−1,p)⊕Hom(Vn−1,p−1,V

∗n−1,p−1).

It follows that⟨⟨x + x′, y + y′⟩⟩ = η⟨x, y⟩n−1,p + αn,p⟨x′, y′⟩n−1,p−1′ .

We can see by setting x′ = y′ = 0 and keeping in mind the matrix Un,p that represents the mapΨ that η = 1, thus proving the lemma.

We keep the ordered basis such that link states with a defect in n come first. In matrix formlemma 4.10 says

Gn−1,p ⊕ αn,pGn−1,p−1 = (U−1n,p)tGn,pU−1

n,p

or in another formulation

Gn,p = UTn,p (Gn−1,p 0

0 αn,pGn−1,p−1)Un,p. (4.4)

Filling in Un,p from equation (4.3) yields

Gn,p = ( Gn−1,p Gn−1,pVn,pV tn,pGn−1,p V tn,pGn−1,pVn,p + αn,pGn−1,p−1

) . (4.5)

The next proposition is crucial for the computation of determinants of Gram matrices.

4.11 Proposition. If [n − 2p + 1]q ≠ 0 (i.e. q is non-critical) and p > 0, we have

αn,p =[n − 2p + 2]q[n − 2p + 1]q

.


Remark. Since both [n − 2p + 2]q and [n − 2p + 1]q are elements of R this yields that αn,p <∞.

Proof. Assume by induction Rn′,p′ = 0 for all n′ < n and for all 0 ≤ p′ ≤ ⌊n2⌋. This holds for

n′ ≤ 2, providing the induction base.We will assume that q is not a root of unity. The cases where q is a root of unity follow from

continuity. We order the basis of Vn,p such that link states with n a defect come first. Thenwe can write G1,1

n,p as the submatrix of Gn,p consisting of products ⟨v,w⟩ where both v and w

have a defect at n. Similarly G1,2n,p consists of ⟨v,w⟩ where v has a defect at n and w does not,

G1,2n,p consists of ⟨v,w⟩ where w has a defect at n and v does not and finally G2,2

n,p consists ofproducts of link states without a defect at n. Thus we can write Gn,p as

Gn,p = (G1,1n,p G1,2

n,p

G2,1n,p G2,2

n,p)

and using equation (4.5) we findG1,2n,p = Gn−1,pVn,p (4.6)

andG2,2n,p = V tn,pGn−1,pVn,p + αn,pGn−1,p−1 = V tn,pG1,2

n,p + αn,pGn−1,p−1. (4.7)

Note that G2,1n,p is the transpose of G1,2

n,p.We make a refinement in the basis, so that link states with n− 1 and n a defect come before

n − 1 part of a link and n a defect, and link states with a link from n − 1 to n come before linkstates with a link in both n − 1 and n, but not between the two. As diagrams this looks asfollows

By definition of the Gram matrix we can make the observation that

G1,2n,p = ( 0 ⋆

Gn−2,p−1 ⋆) and G2,2n,p = (βGn−2,p−1 ⋆

⋆ ⋆) . (4.8)

The top left block of G1,2n,p are forms of link states of type one with type three. They have a link

starting and ending on the same side, so the form becomes zero. As for the bottom left block,the forms stay the same if we cut out the n-th point. The top left block of G2,2

n,p corresponds toforms of two link states of the third type. Cutting away the loop in positions n − 1 and n wecan identify this with Gn−2,p−1, resulting in a factor β.

It is convenient to write

Vn,p = (V1,1n,p ⋆V 2,1n,p ⋆) and Gn−1,p = (G

1,1n−1,p G1,2

n−1,p

G2,1n−1,p G2,2

n−1,p

) .

We may replace G1,1n−1,p in the above expression of Gn−1,p by G1,1

n−2,p because in that blockposition n − 1 carries a defect. Rewrite equation (4.6) as

( 0 ⋆Gn−2,p−1 ⋆) = (G

1,1n−2,p G1,2

n−1,p

G2,1n−1,p G2,2

n−1,p

)(V1,1n,p ⋆V 2,1n,p ⋆) = (G

1,1n−2,pV

1,1n,p +G1,2

n−1,pV2,1n,p ⋆

G2,1n−1,pV

1,1n,p +G2,2

n−1,pV2,1n,p ⋆)


so that we can read

0 = G1,1n−2,pV

1,1n,p +G1,2

n−1,pV2,1n,p (4.9)

Gn−2,p−1 = G2,1n−1,pV

1,1n,p +G2,2

n−1,pV2,1n,p (4.10)

By assumption G1,1n−2,p is invertible. Multiplying (4.9) with (G1,1

n−2,p)−1 we find

0 = V 1,1n,p + (G1,1

n−2,p)−1G1,2n−1,pV

2,1n,p = V 1,1

n,p + Vn−1,pV2,1n,p , (4.11)

where we use (4.6) left-multiplied by (G1,1n−2,p)−1 to obtain the second equality. Next we will

prove

V 2,1n,p = 1

αn−1,p⋅ id . (4.12)

Rewriting equation (4.10) with help of (4.11) gives

Gn−2,p−1 = (G2,2n−1,p −G

2,1n−1,pVn−1,p)V 2,1

n,p . (4.13)

The bottom left block of equation (4.5) with n− 1 instead of n reads G2,1n−1,p = V tn−1,pGn−2,p and

substitution this in equation (4.7) yields

G2,2n−1,p = G

2,1n−1,pVn−1,p + αn−1,pGn−2,p−1,

which in turn impliesαn−1,pGn−2,p−1 = G2,2

n−1,p −G2,1n−1,pVn−1,p. (4.14)

Combining (4.11) and (4.14) shows that

Gn−2,p−1 = Gn−2,p−1 ⋅ (αn−1,pV2,1n−1,p).

By the inductive assumption Gn−2,p−1 is invertible so that the last equation implies (4.12).Next, we look at the top left block of G2,2

n,p as in (4.8). Using what we know, we can rewrite(4.7) as

(βGn−2,p−1 ⋆⋆ ⋆) = ((V

1,1n,p )t (V 2,1

n,p )t⋆ ⋆ )( 0 ⋆

Gn−2,p−1 ⋆) + αn,pGn−1,p−1.

The upper left block of Gn−1,p−1 can be identified with Gn−2,p−1, hence, looking at the upperleft part of the equation only, we find βGn−2,p−1 = (V 2,1

n,p )tGn−2,p−1 + αn,pGn−2,p−1. Since weknow that V 2,1

n,p = 1/αn−1,p ⋅ id, we find the recursive relation

αn,p = β −1

αn−1,p.

One can easily see that α2p,p = β, giving a starting point for the recursion relation.If we set αn,p = [n−2p+2]q

[n−2p+1]qwe see by a computation that αn,p satisfies this recursion relation,

αn,p =q(n−2p+2) − q−(n−2p+2)

q(n−2p+1) − q−(n−2p+1)

= (q + q−1)(q(n−2p+1) − q−(n−2p+1)) − (q(n−2p) − q−(n−2p))q(n−2p+1) − q−(n−2p+1)

= q + q−1 − q(n−2p) − q−(n−2p)

q(n−2p+1) − q−(n−2p+1)

= β − 1

αn−1,p.


If n = 2p we have Gn,p = βGn−1,p−1. In that module Vn−1,p is not defined and (n − 1, p − 1)-linkstates are lifted to Vn,p by connecting the defect to a new point n, thus making an extra link.Using lemma 4.10 we see that Gn,p = αn,pGn−1,p−1 hence α2p,p = β. This starting conditionimplies uniqueness of all αn,p, for we can start with α2p,p and then use the recursion relationto increase 2p in the first index to n.

We can now use equation (4.4) and the fact that detUn,p = 1 to obtain

detGn,p = detGn−1,p ⋅ αdn−1,p−1n,p ⋅ detGn−1,p−1. (4.15)

Here dn−1,p−1 is the dimension of Vn−1,p−1, found in proposition 2.5. We have already seenthat detGn,0 = 1. By substituting G2p,p = βG2p−1,p−1 in the above equation and using α2p,p = βwe find

βd2p−1,p−1 ⋅ detG2p−1,p−1 = det(βG2p−1,p−1) = detGn−1,p ⋅ βdn−1,p−1 ⋅ detGn−1,p−1,

implying detG2p−1,p = 1.The determinant of Gn,p can only be zero if q is a root of unity, hence we find the following

result.

4.12 Corollary. If q is not a root of unity, detGn,p ≠ 0 for all n and p, hence the Vn,p are irreducibleTLn-modules.

In the next section we will find an explicit formula for the determinant detGn,p for all n, pand q (theorem 4.18). To conclude this section, we state semisimplicity of TLn for generic q.

4.13 Theorem. When q is not the root of a unity and β = q + q−1, TL(β) is a semisimple algebra andthe Vn,p form a complete set of non-isomorphic irreducible modules.

Proof. We have seen in proposition 4.12 and 4.5 that the Vn,p are non-isomorphic irreduciblemodules.

Since the number of defects in the upper and lower half of a link state is equal, we haved2n,p = d2

n,p/2 (note that p is even since 2n is). This leads to the identity d2n,n = ∑⌊n/2⌋p=0 d2

n,p, so

dim TLn(β) = d2n,n =⌊n/2⌋

∑p=0

d2n,p =

⌊n/2⌋

∑p=0

(dimVn,p)2

and by Wedderburn’s theorem TL is semisimple.

View TLn as a module over itself (the regular representation). Provided that β is non-zeroand q is not a root of unity, we may now write

TLn(β) ≅⊕p

(dimVn,p) ⋅Vn,p. (4.16)

4.2 The cases q = ±i and q = ±1

When q = ±i, the paramater β equals i+ i−1 = i− i = 0. We will show that TLn(0) is semisimplewhen n is odd. Furthermore, when q = ±1 the Temperley-Lieb algebra is semisimple too. Thisis proven by mimicking techniques from the previous section.

If β = 0 and n > 2p, i.e. the link state has a defect, then the form defined in definition 4.1 isnon-zero, because we can choose link states as in figure 4.2 (left). Hence we can use the same

4.2. The cases q = ±i and q = ±1 47

Figure 4.2: Non-zero form (left) and a single loop (right).

procedure as in section 4.1 to conclude that Vn,p is irreducible.However, when β = 0 and n = 2p, the bilinear form ⟨u, v⟩ = βk = 0 for all u, v ∈ V2p,p, since at

least one loop occurs. Therefore most propositions from section 4.1 do not hold. This problemcan be solved by renormalizing the form.

4.14 Definition. Let n be even. Define the bilinear form [⋅, ⋅] on the TLn(0)-module V2p,p by

[u, v] ∶= limβ→0

⟨u, v⟩β

.

Since ⟨u, v⟩ is of the form βk with k ≥ 1 the form [u, v] is well-defined. It equals 1 if ⟨u, v⟩consists of a single loop and equals 0 otherwise. The new bilinear form inherits bilinearty,symmetry and invariance from the original form ⟨⋅, ⋅⟩. It is nonzero since we can construct asingle loop, like the form [u, v] in the right diagram in figure 4.2 equals limβ→0

ββ= 1.

In order to use proposition 4.5 we formulate an analog for lemma 4.2.

4.15 Lemma. Let u and v be (2p, p − 1)-link states, i.e. link states with two defects. Let u′, v′ ∈ V2p,p

be link states formed by closing the defects. Then for β = 0

∣u, v∣w = [v′,w]u′,

for all w ∈ V2p,p. This extends linearly to all u, v ∈ V2p,p−1.

Proof. On the left hand side, the defects of v are closed by w since w ∈ V2p,p and the result∣u, v∣w equals u′ multiplied by some constant. If no loops occur on the left hand side, then[v′,w] = limβ→0

ββ= 1 so that ∣u, v∣w = u′ = [v′,w]u′. If a loop occurs, both sides become zero

because β = 0 and limβ→00β= 0.

Let R2p,p be the radical of the form [⋅, ⋅],

R2p,p = u ∈ V2p,p ∣ [u, v] = 0 for all v ∈ V2p,p.

Similar to proposition 4.5 we obtain the following result.

4.16 Proposition. If β = 0 and n = 2p the form [⋅, ⋅] is zero and V2p,p is cyclic and indecomposable.Moreover V2p,p/R2p,p.

Proof. The proof is analog to the proof op proposition 4.5.


4.17 Proposition. Every module homomorphism θ ∶ Vn,p → Vn,p is a multiple of the identity.

Proof. For ⟨⋅, ⋅⟩ ≠ 0 this is already proven in proposition 4.6. When ⟨⋅, ⋅⟩ = 0 we must have n = 2pand β = 0. But [⋅, ⋅] ≠ 0. Pick v′,w ∈ V2p,p with [v′,w] = 1. For all u′ ∈ V2p,p let u ∈ V2p,p−1 beobtained from u′ by cutting the link closing at 2p. In particular, construct v in this way. Usinglemma 4.15 we find

θ(u′) = θ(∣u, v∣w) = ∣u, v∣θ(w) = ⟨v′, θ(w)⟩u′,

which proves the proposition.

Remark. The quotient V2p,p/R2p,p is non-zero because the form [⋅, ⋅] is never identically zero,therefore V2p,p ≠ R2p,p.

We can now use the theory from section 4.1 to continue. The main result of this sectionis the following theorem, giving an explicit formula for the determinant of the Gram matrix.The cases q = ±i and q = ±1 will turn up as a corollary.

4.18 Theorem. Let β = q + q−1 ∈ C. Then for all n, p the Gram matrix Gn,p has determinant

detGn,p =p

∏j=1

([n − 2p + 1 + j]q[j]q

)dn,p−j

.

Proof. We have to check that the expression satisfies the recursive relations in equation (4.15)with boundary conditions detGn,0 = 1 and detG2p−1,p = 1.

The first boundary condition is trivial by the convention that the empty product equals 1.For the second one, note that n = 2p − 1 yields [n − 2p + 1 + j]q = [2p − 1 − 2p + 1 + j]q = [j]q ,showing that each factor in the product equals 1, hence so does the product.

As for the relation in (4.15), we need to prove

p

∏j=1

([n − 2p + 1 + j]q[j]q

)dn,p−j

(4.17)

= ([n − 2p + 2]q[n − 2p + 1]q

)dn−1,p−1 p

∏j=1

([n − 2p + j]q[j]q

)dn−1,p−j p−1

∏j=1

([n − 2p + 2 + j]q[j]q

)dn−1,p−1−j

. (4.18)

Using equation (2.1) and the fact that dn−1,p−1−p = 0 we see that

p

∏j=1

( 1

[j]q)dn,p−j

=p

∏j=1

( 1

[j]q)dn−1,p−j+dn−1,p−1−j

=p

∏j=1

( 1

[j]q)dn−1,p−j p−1

∏j=1

( 1

[j]q)dn−1,p−1−j

,

so it suffices to show

p

∏j=1

([n−2p+1+j]q)dn,p−j = ([n − 2p + 2]q[n − 2p + 1]q

)dn−1,p−1 p

∏j=1

([n−2p+j]q)dn−1,p−jp−1

∏j=1

([n−2p+2+j]q)dn−1,p−1−j .

4.2. The cases q = ±i and q = ±1 49

The latter can be done by a direct computation,

([n − 2p + 2]q[n − 2p + 1]q

)dn−1,p−1 p

∏j=1

([n − 2p + j]q)dn−1,p−jp

∏j=1

([n − 2p + 2 + j]q)dn−1,p−1−j

= ([n − 2p + 2]q[n − 2p + 1]q

)dn−1,p−1 p−1

∏j=0

([n − 2p + 1 + j]q)dn−1,p−1−jp

∏j=2

([n − 2p + 1 + j]q)dn−1,p−j

=p−1

∏j=1

([n − 2p + 1 + j]q)dn−1,p−1−jp

∏j=1

([n − 2p + 1 + j]q)dn−1,p−j

=p

∏j=1

([n − 2p + 1 + j]q)dn−1,p−1−jp

∏j=1

([n − 2p + 1 + j]q)dn−1,p−j

=p

∏j=1

([n − 2p + 1 + j]q)dn,p−j .

This proves that the equality in equation (4.18) holds.

4.19 Corollary. If (n, p) is critical, then Rn,p = 0, so Vn,p is irreducible.

Proof. If (n, p) is critical q2(n−2p+1) = 1. Let ` be the smallest integer such that q2` = 1, thenn − 2p + 1 = k` for some k ∈ N and

detGn,p =p

∏j=1

([k` + j]q[j]q

)dn,p−j

.

The numerator vanishes if and only if the denominator vanishes, in which case j = k′`. Wehave

[k`]q =q` − q−`q − q−1

qk` − q−k`q` − q−` = [`]q[k]q` .

Since q` = ±1, [m]q` ≠ 0. For j = k′` we now find

[k` + k′`]q[k′`]q

=[`]q[k + k′]q`[`]q[k′]q`

=[k + k′]q`[k′]q`

≠ 0.

Hence Gn,p ≠ 0.

When q = ±1 we have q2(n−2p+1) = 1 for all pairs (n, p), so every pair is critical. The corollaryabove the implies that TLn(±2) is semisimple. Similarly, when n is odd and q = ±i every pair(n, p) is critical. Hence we obtain the following important corollary.

4.20 Corollary. When β = 2 (q = ±1) the Temperley-Lieb algebra is semisimple. Furthermore, when nis odd and β = 0 (q = ±i), TLn is semisimple.

4.21 Proposition. If β = 0 the radical R2p,p is irreducible.

Proof. Since V2p,p = R2p,p we must consider the radical R2p,p of the form [⋅, ⋅] and the corre-sponding Gram matrix G2p,p. The discussion before corollary 4.12 shows G2p,p = βG2p−1,p−1,so

G2p,p = limβ→0

G2p,p

β= G2p−1,p−1∣β=0

.


When β = 0 we have q = ±i and [k]q = 0 when k is even and ±1 when k is odd. Hence

det G2p,p =p−1

∏j=1

([j + 2]q[j]q

)d2p−1,p−1−j

∣q=±i

.

When j is odd, the factor in the product equals ±1. When j is even, we have j = 2k for somek ∈ N and it follows from [2k]q = [2]q[k]q2 and the fact that q2 = −1 that

[j + 2]q[j]q

= [2(k + 1)]q[2k]q

=[k + 1]q2[k]q2

≠ 0

This implies det G2p,p ≠ 0 so that R2p,p is irreducible.

4.3 Roots of unity

In this section we study representations of TLn(β), where β = q+q−1 and q ∈ C is a root of unity.Our aim is to give the principle indecomposable modules of the Temperley-Lieb algebra. Atthe end of the section, in theorem 4.32, it is proven that the principle indecomposables eithercoincide with the standard modules Vn,p or with the direct summands of the k-fold inducedmodule of the standard module.

4.3.1. Bratelli diagrams

By counting basis elements, we know dimVn,p = dimVn−1,p + dimVn−1,p−1. Let q be a rootof unity and let ` be minimal such that q2` = 1. Given (n, p) set kn,p, rn,p ∈ N such thatn − 2p + 1 = kn,p ⋅ ` + rn,p, with 0 < rn,p ≤ `. Then if q is a root of unity, (n, p) is critical if andonly if rn,p = `. Using this, we can make the following statement about the dimension of theradical Rn,p.

4.22 Proposition. Let q be a root of unity. The dimension of the radicals Rn,p satisfies

dimRn,p =⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if rn,p = `dimRn−1,p + dimVn−1,p−1 if rn,p = ` − 1dimRn−1,p + dimRn−1,p−1 otherwise

,

with initial conditions dimRn,0 = 0 and dimR2p−1,p = 0.

Proof. Since for all n Gn,p = (1) and for all p the module V2p−1,p is not defined, we havedimRn,0 = 0 and dimR2p−1,p = 0. If rn,p = ` the couple (n, p) is critical and by corollary 4.19

dimRn,p is zero.Now assume rn,p ≠ `, then q2(n−2p+1) = q2kn,p`+2rn,p = q2rn,p ≠ 1 since ` is minimal. The

dimension of Rn,p equals the dimension of the kernel of Gn,p, so this we will compute. Byproposition 4.9 the exact sequence splits, consequently we may rewrite Gn,p as in equation(4.4),

Gn,p = UTn,p (Gn−1,p 0

0 αn,pGn−1,p−1)Un,p.

As Un,p is invertible, Gn,pv = 0 if and only if Gn−1,pw1 = αn,pGn−1,p−1w2 = 0 where (w1,w2)T =Un,pv. When r(n, p) = ` − 1, [n − 2p + 2]q = 0 so that αn,p = 0 by proposition 4.11 and kerGn,p =kerGn−1,p ⊕ Vn−1,p−1. If r(n, p) ∈ 1,2, . . . , ` − 2 we find αn,p ≠ 0 by the same proposition,resulting in kerGn,p = kerGn−1,p ⊕ kerGn−1,p−1.

4.3. Roots of unity 51

We can immediately extract an analogous result for the irreducible quotients Ln,p = Vn,p/Rn,p.Namely:

4.23 Corollary. The dimension of Ln,p satisfies

dimLn,p =⎧⎪⎪⎪⎨⎪⎪⎪⎩

dimVn,p if rn,p = `dimLn−1,p if rn,p = ` − 1dimLn−1,p + dimLn−1,p−1 otherwise

,

with initial conditions dimLn,0 = 1 and dimL2p−1,p = 0.

Using the previous proposition and corollary, we can make tables of dimensions of radicalsRn,p and irreducibles Ln,p for varying n and p. Arrange the tables such that n is constant onthe horizontal lines and p is constant on the diagonal lines as in figure 4.3. Such tables arecalled Bratelli diagrams.

(8,4) (8,3) (8,2) (8,1)(7,3) (7,2) (7,1)

(6,3) (6,2) (6,1) (6,0)(5,2) (5,1) (5,0)

(4,2) (4,1) (4,0)(3,1) (3,0)

(2,1) (2,0)(1,0)

Figure 4.3: Arrangement of pairs in a Bratelli diagram with critical lines for q ≠ 1 a root ofunity with q5 = 1, thus ` = 5. The red circles indicate the orbit of (6,3).

4.24 Definitions. Since (n, p) critical implies that (n+2, p+1) is critical, the critical pairs (n, p)form vertical lines in the Bratelli diagram, called critical lines.

We call a pair (n, p′), (n, p) with 0 < ∣p′ − p∣ < ` symmetric if they are both non-critical and ifthey are located symmetrically on each side of a single critical line.

0

0 0

0 0

0 0 0

0 1 0

0 1 0 0

1 6 0 0

1 7 0 0 0

8 27 0 0 0

8 35 0 0 1 0

43 110 0 1 0 0

1

1 1

2 1

2 3 1

5 3 1

5 8 5 1

13 8 6 1

13 21 20 7 1

34 21 27 8 1

34 55 75 35 8 1

89 55 110 43 10 1

Figure 4.4: Bratelli diagrams with dimRn,p (left) and dimLn,p (right) for q a primitive fifthroot of unity, thus ` = 5.


It is natural to consider orbits of p around reflections of the critical lines. Every pair (n, p)belongs to exactly one orbit. Let p1 be an element to the left of the left-most element of anorbit. By reflection we obtain p2, p3, . . . with p1 > p2 > p3 > ⋯ who form an orbit. If pi andpi+1 are in the same orbit and are separated by one critical line then kn,pi+1 = kn,pi + 1 andrn,pi+1 = ` − rn,pi . By the identity n − 2p + 1 = kn,p` + rn,p we find

pi+1 = pi + rn,pi − ` and pi−1 = pi + rn,p′ . (4.19)

Furthermore gn,pi−1 = gn,pi , where gn,p denotes the eigenvalue of the critical element Jn (cf.proposition A.1). This can be seen by computing

gn,pi−1 = (−1)n(qn−2pi−1+1 + q−(n−2pi−1+1))= (−1)n(qkn,pi−1`+rn,pi−1 + q−kn,pi−1`−rn,pi−1 )= (−1)n(q(kn,pi

−1)`+`−rn,pi + q−(kn,pi−1)`−`+rn,pi )

= (−1)n(qkn,pi`−rn,pi + q−kn,pi

`+rn,pi )= (−1)n(q−2kn,pi

`qkn,pi`−rn,pi + q2kn,pi

`q−kn,pi`+rn,pi )

= gn,pi . (4.20)

It follows that gn,p = gn,p′ for all p, p′ in the same orbit.When β = 0 with n odd or β = ±2 every (n, p) is critical. Therefore there are no symmetric

pairs in these cases. In all other cases such pairs exist whenever n > `. There appears tobe a connection to the dimension of certain radicals to the dimension of irreducibles. Thefollowing propositions makes this precise.

4.25 Proposition. Given a root of unity q and a non-critical pair (n, p). Then

dimRn,p = dimLn,p+r(n,p)−`

if p + r(n, p) − ` ≥ 0 and dimRn,p = 0 otherwise.

Note that (n, p) and (n, p + rn,p − `) form a symmetrical pair.

Proof. When n = 1, p+r(n, p)−` = 2−` < 0 since (n, p) non-critical implies ` > 2, so dimRn,p = 0as needed.

As induction hypothesis, assume that the proposition is true for n−1 and all p. Both (n−1, p)and (n − 1, p − 1) may or may not be critical, resulting in four cases.

Case 1. Both (n − 1, p) and (n − 1, p − 1) are non-critical. Since we can write (n − 1) − 2p + 1 =k(n− 1, p)`+ r(n− 1, p) = k(n, p)`+ r(n, p)− 1 we have r(n, p)− 1 = r(n− 1, p) ≠ ` so r(n, p) ≠ 1.Similarly (n−1)−2(p−1)+1 = k(n, p)`+ r(n, p)+1 implies r(n, p) is not equal to `−1. Besides(n, p) is non-critical, so r(n, p) ∉ 1, ` − 1, `. By the induction hypothesis

dimRn−1,p = dimLn−1,p+r(n−1,p)−` = dimLn−1,p−1+r(n,p)−` (4.21)

anddimRn−1,p−1 = dimLn−1,p−1+r(n−1,p−1)−` = dimLn−1,p+r(n,p)−`. (4.22)

By proposition 4.22 dimRn,p = dimRn−1,p + dimRn−1,p−1. Direct computation gives r(n, p +r(n, p) − `) = −r(n, p) modulo `, hence r(n, p + r(n, p) − `) ∉ ` − 1, ` and by corollary 4.23

dimLn,p+r(n,p)−` = dimLn−1,p+r(n,p)−`+dimLn−1,p−1+r(n,p)−`, the sum of the right hand sides ofequations (4.21) and (4.22).


Case 2. Only (n − 1, p − 1) is critical. Then ` = r(n − 1, p − 1) = r(n, p) + 1 so r(n, p) = ` − 1.Equation (4.21) still holds. Instead of (4.22) we get

dimVn−1,p−1 = dimLn−1,p−1 = dimLn−1,p+r(n,p)−` (4.23)

by corollary 4.19. Applying proposition 4.22 and corollary 4.23 again yields the result.

Case 3. Only (n − 1, p) is critical. Now r(n, p) = 1 and equation (4.22) is valid. The fact thatdimRn−1,p = 0 provides the result.

Case 4. Both (n − 1, p) and (n − 1, p − 1) are critical. Now r(n, p) = 1 = ` − 1 thus ` = 2. Theresult follows from combining equation (4.23) and dimRn−1,p = 0.

This subsection has investigated the dimensions of radicals and irreducibles of standardmodules when q is a root of unity. This will play a vital role in the description of the principleindecomposable modules of TLn in subsection 4.3.3.

4.3.2. Irreducibility of the radicals

This subsection is concerned with the irreducibility of the radicals of the standard modules(theorem 4.28). The proof uses a nonzero map Vn,p → Vn,p′ for certain p and p′ constructed inlemma 4.27 based on the nontrivial action of Jn+1 on Vn,p↑. The next lemma states that theaction of the central element Jn+1 on Vn,p↑ is nontrivial.

Consider the element Jn+1(1⊗ zp) in Vn,p↑. Then we can write it as a sum of basis elementsof Vn,p↑ (see corollary cor-ind-module-basis) multiplied by some coefficient. One of the termsin the sum is e1e2⋯en ⊗ zp = e2p+1e2p+2⋯en ⊗ zp. Let α denote its coefficient. The followinglemma claims that, when q is a root of unity and under some mild conditions, the coefficientα is not equal to 0.

4.26 Lemma. Let q ≠ ±1 be a root of unity and let (n, p) be critical with respect to this q (so n ≠ 2p).Let zp denote the (n, p)-link state which has p simple links at 1,3, . . . ,2p−1. Then when Jn+1(1⊗ zp)is expanded in the basis In,p of Vn,p↑, the coefficient of e1e2⋯en⊗ zp = e2p+1e2p+2⋯en⊗ zp is nonzero.

Proof. We devide the prove into three cases. In the first case, we assume p = 0. In the secondcase this is used to prove the statement for p ≥ 1 and β ≠ 0. The last case covers β = 0.

(Case 1) Set p = 0. Since Jn+1 ∈ TLn+1 it can be written as a sum of words in reverse Jones’normal form, cf proposition 2.8. Such a word acts in a nontrivial way on 1 ⊗ z0 only if therightmost letter is en. So the only word that will contribute to the term e1e2⋯en ⊗ z0 we arelooking for is e1e2⋯en. Thus we want to know the coefficient of the term e1e2⋯en in Jn+1.

One may draw the diagram e1e2⋯en. It is equal to the diagram in figure 4.5.

Figure 4.5: The diagram e1e2⋯en.


In Jn, the vertices 1 and 2 need to be linked in order to “become” e1e2⋯en. Expanding thecrossings in the first row yields the equation in figure 4.6.

1

2= − q−1 + −q

Figure 4.6: Computation.

In the last term 1 cannot be connected to 2, hence it does not contribute to the word. Leavingit out leaves us with the left diagram in figure 4.7 to resume our computation. On the right ofthe second row, only one tile is possible, for it may not close a link on the right line. This leavesthe left tile of the second row with one possibility to add to e1e2⋯en. The same argument canbe repeated for all but the last row. The result is given in the right diagram of figure 4.7. Thefactor stays the same since each row contributes a factor 1.

(2 − βq−1) ⋅ (2 − βq−1) ⋅

1

2

n − 1n

Figure 4.7: Computation.

The bottom row can be studied the same way as the top row, resulting in a factor 2 − qβ.Thus the coefficient of e1e2⋯en in Jn+1 equals

(2 − βq−1)(2 − βq) = 4 − β2

and vanishes only when β = ±2 (hence only when q = ±1).

(Case 2) Let p ≥ 1 and β ≠ 0. The link state zp then has a simple link at 1. Let v′ be a linkstate in Vn,p with a simple link at 1. Then 1⊗w = β−1e1(1⊗ v′). Expand the top two crossings


in the right column of Jn+1:

Jn+1(1⊗w) = ×⎛⎜⎝c2 + + c−2

⎞⎟⎠

(1⊗w)

= ×⎛⎜⎝β−1c2 + + β−1c−2

⎞⎟⎠

(1⊗w)

= ×⎛⎜⎜⎝(β−1(c2 + c−2)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

=−1

+1) +⎞⎟⎟⎠

(1⊗w)

= (1⊗w).

Next we expand the top two crossings of the left column. This yields

Jn+1(1⊗w) =

⎛⎜⎜⎜⎜⎜⎝

c2 + + + c−2

⎞⎟⎟⎟⎟⎟⎠

(1⊗w)

= β−1 (1⊗w) = β−1 (1⊗w) = (1⊗w).

In the second equality, the first, second and fourth diagram cancel. In the fourth equality weuse that β−1e1(1⊗w) = 1⊗w.

If w happens to have another simple link at 3, the same procedure can be used. In general,for zp we have

Jn+1(1⊗ zp) =

Jn−2p+1

12

2p (1⊗w).

If n > 2p, we know from the case p = 0 that the coefficient of e2p+1e2p+2⋯en is 4 − β2, so that,under the assumption, the coefficient of e2p+1e2p+2⋯en ⊗ zp ≠ 0. If n = 2p, the coefficient is 1.

(Case 3) Assume β = 0, which corresponds to q = i or q = −i. The element Jn is a linearcombination of words in e1, . . . , en−1 with weights polynomial in q and q−1. Note that

ei(er⋯en ⊗w) =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

er⋯en ⊗ eiw i < r − 1er−1⋯en ⊗w i = r − 1βer⋯en ⊗w i = rei⋯en ⊗ er⋯ei−2w r + 1 ≤ i ≤ n

. (4.24)


The computation in (4.24) shows that coefficients of Jn+1 is the basis In,p of Vn,p ↑ are alsopolynomial in q and q−1. Therefore, the value at β = 0 can be found by evaluating β = 0in the polynomial 4 − β2, found in the previous steps, hence the coefficient of e1⋯en ⊗ zp inJn+1(1⊗ zp) equals 4 − 02 = 4 and is nonzero.

4.27 Lemma. Let q be a root of unity with ` ≥ 3 or ` = 2 and n even. Let Vn,p,Vn,p′ be a symmetricalpair with p′ > p. Then there exists a non-zero homomorphism Vn,p → Vn,p′ .

Proof. Assume (n, p) is critical, q is a root of unity and ` ≥ 3 or ` = 2 with n is even. Then thepair (n + 1, p), (n + 1, p + 1) is symmetric and

gn+1,p = gn+1,p+1.

Define φ ∶ Vn,p↑→ Vn,p↑∶ x ↦ (Jn+1 − gn+1,p1)x. Since Jn+1 − gn+1,p1 is central we have xφ(w) =x(Jn+1 − gn+1,p1)w = (Jn+1 − gn+1,p1)xw = φ(xw), so φ is a module homomorphism. Besides, φis nonzero by lemma 4.26.

Recall the short exact sequence 0→ Vn+1,p+1αÐ→ Vn,p↑

γÐ→ Vn+1,p → 0 from corollary 3.12. Byconstruction Vn+1,p+1 is mapped to 0 under φ, so that imα ⊆ kerφ. Besides γ φ = 0 whereforeimφ ⊆ kerγ = imα. Let w ∈ Vn+1,p, then there exists v ∈ Vn,p↑ such that γ(v) = w. Supposev′ ∈ Vn,p is such that γ(v′) = w, then v − v′ ∈ kerγ ⊆ kerφ, so the map w ↦ φ(v) is well-defined. This is a homomorphism since φ(xv) = xφ(v). Define the map κ ∶ Vn+1,p → Vn+1,p+1

by w ↦ (α−1 φ)(v). This is nonzero because φ is nonzero and α is injective (so α−1 of anonzero element is nonzero). Therefore dim HomTLn+1(Vn+1,p,Vn+1,p+1) ≥ 1.

When (n, p) is critical the pair Vn+j,p and Vn+j,p+j is symmetric for all 1 ≤ j < `. On the otherhand, every symmetric pair is of this form. (This is clear from figure 4.3). We claim that

HomTLn+j(Vn+j,p,Vn+j,p+j) = HomTLn+1(Vn+1,p,Vn+1,p+1) ≠ 0.

The latter equality we have already seen. The first equality follows from the following,

HomTLn+j(Vn+j,p,Vn+j,p+j) = HomTLn+j(Vn+j,p ⊕Vn+j,p+1,Vn+j,p+j)= HomTLn+j(Vn+j−1,p↑,Vn+j,p+j)= HomTLn+j−1(Vn+j−1,p,Vn+j,p+j↓)= HomTLn+j−1(Vn+j−1,p,Vn+j−1,p+j ⊕Vn+j−1,p+j−1)= HomTLn+j−1(Vn+j−1,p,Vn+j−1,p+j−1).

The first equality holds because gn+j,p+1 ≠ gn+j,p+j hence HomTLn+j(Vn+j,p+1,Vn+j,p+j) = 0 and

HomTLn+j(Vn+j,p,Vn+j,p+j)⊕HomTLn+j(Vn+j,p+1,Vn+j,p+j) = HomTLn+j(Vn+j,p⊕Vn+j,p+1,Vn+j,p+j).

The last equality is dealt with in a similar way. The second and fourth equality are an appli-cation of the splitting of exact sequence (proposition 4.9) and the fact that (n + j, p + j) and(n + j + 1, p + 1) are non-critical. The third one is an application of Frobenius reciprocity, cf.proposition A.6. It follows that there exists a non-zero homomorphism Vn,p → Vn,p′ for everysymmetric pair with p′ > p.

4.28 Theorem. The radical Rn,p is zero or irreducible. Moreover if Vn,p and Vn,p′ is a symmetric pairwith p′ > p, we have

Rn,p′ ≅ Ln,p.


Proof. When q is not a root of unity corollary 4.12 states Rn,p = 0. For n odd and q = ±1 or ±ithe same holds by applying theorem 4.18. Assume (n, p) is a root of unity and ` ≥ 3 or ` = 2and n is even.

Let f ∈ Hom(Vn,p,Vn,p′) (p′ > p) be a non-zero homomorphism which exists by lemma 4.27.Then ker f ⊊ Vn,p, so by maximality of the radical ker f is a subset of Rn,p. Suppose f issurjective, then Vn,p/ker f ≅ Vn,p′ which is in contradiction with proposition 4.4. Hence againby maximality of the radical we find im f ⊆ Rn,p′ . If either ker f or im f is a proper subset ofthe radical we would have dim ker f < dimRn,p or dim im f < dimRn,p′ so that

dimVn,p = dim ker f + dim im f < dimRn,p + dimRn,p′ .

This in turn impliesdimLn,p = dimVn,p − dimRn,p < dimRn,p′ ,

which is in contradiction with proposition 4.25. Thus ker f = Rn,p and im f = Rn,p′ and by thefirst isomorphism theorem

Ln,p = Vn,p/ker f ≅ im f = Rn,p′ .

If β = 0 and n = 2p′ then by proposition 4.21 R2p′,p′ = V2p′,p′ is irreducible and since f isnon-zero im f = R2p′,p′ . The remainder of the proof is analogous.

If the reflection of (n, p′) in the line immediately to its right is well-defined, the argumentproves above proves irreducibility of Rn,p′ . If such a reflection is not well-defined (for exampleit is not well-defined for the pair (4,1) in figure 4.3) then the radical equals zero by proposition4.25.

4.29 Corollary. Every quotient of a standard module is indecomposable.

Proof. If β = 0 the module V2p,p is irreducible and the claim is immediate. In all other cases, theonly proper non-zero submodule of Vn,p is its radical, hence the only quotients are 0,Ln,pand Vn,p. The module Ln,p is indecomposable because it is isomorphic to Rn,p′ for somep′ and Rn,p′ is irreducible hence indecomposable. The module Vn,p is indecomposable byproposition 4.5

4.30 Corollary. Exclude V2p,p when β = 0 (so assume either n > 2p or β ≠ 0). The standard moduleVn,p is reducible if and only if (n, p) forms a symmetric pair with (n, p′) where p > p′. Then thesequence

0 Ln,p′ Vn,p↑ Ln,p 0. (4.25)

is exact and does not split. Therefor, if Vn,p is reducible it has two composition factors, Ln,p′ and Ln,p,and the composition series is 0 ⊆ Ln,p′ ⊆ Vn,p.

Proof. The module Vn,p is reducible if and only if its submodule Rn,p is non-zero, whichhappens if and only if (n, p) forms a symmetric pair with (n, p′) and p > p′ by theorem 4.28.

Noting that Rn,p ≅ Ln,p′ , it is trivial that the sequence is exact. The module is indecompos-able by proposition 4.3, hence the short exact sequence cannot split.

It is clear that 0 ⊆ Ln,p′ ⊆ Vn,p is the composition series by definition.

For completeness, we state the following theorem. The result is not used in the remainderof the thesis, but nevertheless is interesting in itself. For a proof we refer to theorem 7.5 of[26].

4.31 Theorem. The dimension dim Hom(Vn,p,Vn,p′) = 1 if p = p′ or if the two standard modules forma symmetric pair with p′ > p. There is a single exceptional case: if β = 0 dim Hom(V2,1,V2,0) = 1.Otherwise dim Hom(Vn,p,Vn,p′) = 0.


4.3.3. Principal indecomposable modules

In this subsection we will give an explicit description of the principal indecomposable mod-ules. These are the indecomposable direct summands of the Temperley-Lieb algebra whenviewed as a module over itself. There is a bijective correspondence between the principle in-decomposables and the irreducibles by quotienting out the a principle indecomposable mod-ule by its unique maximal proper submodule. We will denote the principle indecomposablecorresponding to the irreducible module Ln,p by Pn,p.

Obviously, when TLn is semisimple, Pn,p = Vn,p = Ln,p. Since TLn(β) is semisimple forgeneric β (independent of n) it follows from theorem A.3 and A.5 that

∑i

dimLi dimPi = dim TLn =∑p

(dimVn,p)2.

There is no guarantee yet that the Ln,p form a complete set of irreducibles. This will beproven as a corollary of the following theorem, which formulates the main result of thissection.

4.32 Theorem. Let q be a root of unity and ` the minimal positive integer such that q2` = 1. Recalln − 2p + 1 = kn,p` + rn,p. Then the principal indecomposables are as follows.

• If rn,p = ` (so (n, p) is critical), Pn,p ≅ Vn,p.

• If kn,p = 0 (so n − 2p + 1 ≤ `) and β ≠ 0 then Pn,p ≅ Vn,p.

• If kn,p > 0 and rn,p ≠ ` then Pn,p is the direct summand of the rn,p-fold induced moduleVn−rn,p,p ↑ ⋯↑ (rn,p arrows), consisting of the generalised eigenspace under the action of Jn,whose generalised eigenvalue is gn,p. Also, there is a short exact sequence

0 Vn,p+rn,p Pn,p Vn,p 0 (4.26)

which does not split.

Besides, the set Ln,p (with p ∈ 0,1, . . . , ⌊n2⌋) is a complete set of pairwise non-isomorphic irre-

ducible modules.

Let us start with some easy special cases. These will deal with the event ` ≤ 2 and they willserve as the inductive step in the proof of the theorem.

4.33 Proposition. The theorem holds when n = 1.

Proof. The proof of this is left to the reader.

4.34 Proposition. When q = ±1 (so β = ±2 and ` = 1) or q = ±i (β = 0, ` = 2) and n is odd, we havePn,p ≅ Vn,p ≅ Ln,p for p = 0,1, . . . , ⌊n

2⌋ and the Ln,p form a complete set of pairwise non-isomorphic

irreducible modules.

Proof. In this case, TLn(β) is semisimple and the result is immediate.

4.35 Proposition. If q = ±i (so β = 0 and ` = 2) and n is even, we have Pn,p ≅ Vn−1,p ↑ for p =0,1, . . . , n

2− 1 and for those p the Ln,p form a complete set of pairwise non-isomorphic irreducible

modules.


Proof. Since TLn−1 is semisimple, Wedderburn’s theorem tells us

TLn−1 ≅n/2−1

⊕p=0

(dimLn−1,p)Vn−1,p.

Since TLn = TLn−1↑ (as left modules over TLn), it follows that

TLn ≅n/2−1

⊕p=0

(dimLn−1,p)Vn−1,p↑ =n/2−1

⊕p=0

(dimLn,p)Vn−1,p↑, (4.27)

where we have used that rn,p = 1 = ` − 1 so that corollary 4.23 implies dimLn−1,p = dimLn,p.Recall the short exact sequence 0 → Vn,p+1 → Vn−1,p↑→ Vn,p → 0 (corollary 3.12). The sequenceimplies that Ln,p is an irreducible quotient of Vn−1,p↑, hence Pn,p is one of the indecomposablesthat Vn−1,p ↑ is made of, so Vn−1,p ≅ Pn,p ⊕ . . . It follows from the sequence and the factthat all composition factors of the standard modules are of the form Ln,p′ that any otherprincipal indecomposable is of the form Pn,p′ . By theorem A.5 the multiplicity of Pn,p in TLnis dimLn,p, so the only way to be consistent with equation (4.27) is to set Vn−1,p ↑≅ Pn,p.

Now we have handled the cases where ` ≤ 2 or n = 1. In the following we will assume ` > 2and n > 1. Let us proceed with a general proposition.

Assumption. We will now make the inductive assumption that theorem 4.32 holds for alln′ < n.

4.36 Proposition. The principal indecomposables of TLn are of the form Pn,p.

Proof. We have

TLn =⌊(n−1)/2⌋

⊕p=0

(dimLn−1,p)Pn−1,p↑ .

On the other hand, theorem A.5 gives

TLn =⌊n/2⌋

⊕p=0

(dimLn,p)Pn,p ⊕Pnew,

where Pnew is not one of the Pn,p′ . To prove the proposition we have to show that Pnew = 0.We consider three cases.

Case 1. If kn−1,p = 0 and rn−1,p < `, then by induction Pn−1,p ≅ Vn−1,p and Pn−1,p↑ ≅ Vn−1,p↑≅ Vn,p+1 ⊕ Vn,p since the sequence from corollary 3.12 splits. We identify Pn,p+1 ≅ Vn,p+1 andPn,p ≅ Vn,p (indecomposable by proposition 4.3) and find

Pn−1,p↑ ≅ Pn,p+1 ⊕Pn,p. (4.28)

We conclude Pnew = 0.

Case 2. If rn−1,p = ` the pair (n − 1, p) is critical. Again we have Pn−1,p ≅ Vn−1,p by theinductive assumption. The exact sequence 0 → Vn,p+1 → Pn−1,p↑→ Vn,p → 0 does not split, butit does tell that Ln,p is a quotient of Pn−1,p↑, so Pn−1,p↑≅ Pn,p ⊕ P′. Here P′ is zero or it is thedirect sum of principal indecomposables of the form Pn,p′ , because the irreducible factors ofthe terms of Pn−1,p↑ are those of Vn,p+1 and Vn,p, which are Ln,p′ for some p′. Again, we mayconclude Pnew = 0.


Case 3. Otherwise kn−1,p > 0 and rn−1,p < `. By the induction hypothesis we have a shortexact sequence 0→ Vn−1,p+rn−1,p → Pn−1,p → Vn−1,p → 0. By A.7 the sequence

Vn−1,p+rn−1,p↑ Pn−1,p↑ Vn−1,p↑ 0

is exact and since the pairs (n − 1, p + rn−1,p) and (n − 1, p) are non-critical, proposition 3.6entails the exactness of

Vn,p+rn−1,p ⊕Vn,p+rn−1,p+1 Pn−1,p↑ Vn,p ⊕Vn,p+1↑ 0.

Projecting this sequence onto the generalised eigenspaces of Jn yields the sequences

Vn,p+rn−1,p+1 P Vn,p 0 and Vn,p+rn−1,p P′ Vn,p+1 0,

where Pn−1,p↑ = P ⊕ P′. Since P has irreducible quotient Ln,p we have P = Pn,p ⊕ .. SimilarlyP′ = Pn,p+1 ⊕ P′old so that Pn−1,p↑ is the direct sum of indecomposables of the form Pn,p′ . Wemay conclude Pnew = 0.

In all cases we found Pnew = 0, hence we conclude that all indecomposable modules ofTLn(β) are of the form Pn,p.

4.37 Corollary. Let n > 1. Let q be a root of unity and 2 < ` ∈ N minimal such that q2` = 1. Then theset Ln,p with p = 0,1, . . . , ⌊n

2⌋ is a complete set of non-isomorphic irreducible TLn(β)-modules.

Proof. We know that the principal indecomposables of TLn are Pn,p (p = 0,1, . . . ⌊n2⌋). Their

irreducible quotients form a complete set of irreducible quotients and are precisely the Ln,p.

Using the inductive assumption made above, let us now prove the theorem.

Proof of theorem 4.32. Corollary 4.37 proves that the Ln,p form a complete set of non-isomorphicirreducibles of TLn. Thus we have to prove the description of Pn,p and that the sequence in(4.26) is exact.

Case 1. Let 1 < rn,p < ` − 1. Then (n − 1, p) and (n − 1, p − 1) are non-critical.If kn,p = 0, we are in case 1 of proposition 4.36 and by equation (4.28) we know that each

copy of Pn−1,p ↑ and Pn−1,p−1 ↑ contributes to one copy of Pn,p to TLn. This adds up todimLn−1,p + dimLn−1,p−1 = dimLn,p copies of Pn,p (we used corollary 4.23) and completelyaccounts for the multiplicity of Pn,p. That is, Pn,p may not occur in any of the Pold’s for1 < rn,p < ` − 1. Besides, case 1 of the proposition gives Pn,p ≅ Vn,p, which was to be proven inthe theorem.

If kn,p > 0 we are in case 3 of proposition 4.36. The exact sequences

Vn,p+rn−1,p+1 P Vn,p 0 and Vn,p+rn−1,p−1−1 P′ Vn,p 0,η η′

imply that P and P′ each have at least one Pn,p in the summand, hence Pn−1,p↑ and Pn−1,p−1↑contribute at least dimLn−1,p + dimLn−1,p−1 = dimLn,p copies of Pn,p. If P /≅ Pn,p or P′ /≅ Pn,p,then we could get additional copies of Pn,p′ with eigenvalue gn,p′ = gn,p (the equality wasshown in equation (4.20)). Such copies have 1 < rn,p < ` − 1, hence they are already covered.Since we cannot have extra copies of Pn,p′ , we must have P ≅ Pn,p and P′ ≅ Pn,p.

Note that p + rn−1,p + 1 = p + rn−1,p−1 − 1 = p + rn,p. The above sequences entail the followingshort exact sequences,


0Vn,p+rn,p

kerηP Vn,p 0

and

0Vn,p+rn,p

kerη′P′ Vn,p 0.

We will show at the end of this proof that kerη = 0 = kerη′ to find the sequence given in thetheorem.

Case 2. Let rn,p = 1 or `− 1. If rn,p = 1 the pair (n− 1, p) is critical and (n− 1, p− 1) is not (for` > 2 by assumption). Using case 2 of proposition 4.36 we can make the same analysis as incase 1. Be that as it may, additional Pn,p′ with fn,p′ = fn,p may have rn,p′ = ` − 1. This leads tothe second part of the case, where we will prove that then also, all Pn,p′ are accounted for.

So assume rn,p = ` − 1. Then (n − 1, p) is non-critical and (n − 1, p − 1) is critical. As in case1, inducing Pn−1,p entails the short exact sequence

0Vn,p+rn,p

kerηP Vn,p 0

and hence Pn−1,p↑ contributes dimLn−1,p copies of Pn,p to TLn. Corollary 4.23 gives dimLn−1,p =dimLn,p, so all copies of Pn,p are already accounted for solely by inducing Pn−1,p. The sameargument as in case 1 shows that P = Pn,p for rn,p = 1 or `−1. It remains to show that kerη = 0,which is done after case 3.

Case 3. Suppose rn,p = `. Then rn−1,p = ` − 1 and rn−1,p−1 = 1. We are looking for multiplicitydimLn,p for the principal indecomposable module Pn,p. First compute

dimLn,p = dimVn,p

= dimVn−1,p + dimVn−1,p−1

= dimLn−1,p + dimRn−1,p + dimLn−1,p−1 + dimRn−1,p−1

= dimLn−1,p + 2 dimLn−1,p−1 + dimLn−1,p−`, (4.29)

where the second equation holds by proposition 4.9, the third by propostion 4.3 and the fourthby proposition 4.25.

Inducing and projecting modules as above, we find exact sequences for the summands P ofPn−1,p↑ and P′ of Pn−1,p−1↑, whose Fn-eigenvalue is fn,p,

Vn,p+` P Vn,p 0 and Vn,p P′ Vn,p 0.η η′

The standard modules in these sequences are all critical, hence by corollary 4.19 irreducible.There are three possibilities for P:

(i) kerη = Vn,p+`, then P ≅ Vn,p and each Pn−1,p↑ contributes one copy of Pn,p for a total ofdimLn−1,p copies.

(ii) kerη = 0 and P is indecomposable with exact sequence 0 → Vn,p+` → P → Vn,p → 0. Nowtoo, each Pn−1,p↑ contributes one copy of Pn,p for a total of dimLn−1,p copies.

(iii) kerη = 0 and P decomposes as Vn,p+`⊕Vn,p. In this case Pn−1,p↑ contributes one Pn,p andone Pn,p+` and we find a total of dimLn−1,p + dimLn−1,p−` copies of Pn,p.


In a similar way we find the following three cases for P′, either (i) P′ ≅ Vn,p, (ii) P′ is indecom-posable with exact sequence 0→ Vn,p → P′ → Vn,p → 0, or (iii) P′ ≅ 2Vn,p.

In order to reach the multiplicity of Pn,p found in (4.29) we must have the third possibilityfor both P and P′. Thus P ≅ Vn,p+`⊕Vn,p and P′ ≅ 2Vn,p. It follows that Pn,p ≅ Vn,p, as desired.

The short exact sequences. We already know that the sequence

0Vn,p+rn,p

kerηP Vn,p 0 (4.30)

from case 1 and 2 is exact. In order to prove that the exact sequence (4.26) is exact, it sufficesto prove that kerη is trivial. The equation still holds when kn,p = 0 if we understand Vn,p withp > n

2to be zero.

We have seen⌊n/2⌋

∑p=0

dimLn,p dimPn,p =⌊n/2⌋

∑p=0

(dimVn,p)2. (4.31)

If (n, p) is critical Pn,p ≅ Vn,p ≅ Ln,p so kerη and kerη′ vanish. For non-critical (n, p), we cansplit the sum in a sum over orbits under reflection about critical lines (cf. definition 4.24),where the orbit p1 > p2 > ⋯ > pm ≥ 0 is indexed by p1. This gives

∑n−2p1+1<`

m

∑i=1

dimLn,pi dimPn,pi = ∑n−2p1+1<`

m

∑i=1

(dimVn,pi)2.

Using the exact sequence in (4.30) and the relations in equation (4.19) we find the bound

dimPn,pi ≤ dimVn,pi+rn,pi+ dimVn,pi = dimVn,pi−1 + dimVn,pi . (4.32)

(Modules indexed by p0 or pm+1 are understood the be zero.) In (4.32) equality holds if andonly if kerη = 0. The inequality yields a larger inequality:

m

∑i=1

dimLn,pi dimPn,pi ≤m

∑i=1

(dimLn,pi dimVn,pi + dimLn,pi dimVn,pi−1) (4.33)

and by using proposition 4.3 and proposition 4.25 (in that order) it is seen to equal

=m

∑i=1

((dimVn,pi)2 − dimRn,pi dimVn,pi + dimLn,pi dimVn,pi−1)

=m

∑i=1

((dimVn,pi)2 − dimLn,pi+1 dimVn,pi + dimLn,pi dimVn,pi−1)

=m

∑i=1

(dimVn,pi)2.

By equation (4.31) we know that (4.33) should be an equality, which is the case only if kerη =kerη′ = 0. This proves that the sequence in the theorem is short exact.

Chapter 5

Examples in statistical physics

This chapter shows the use of the Temperley-Lieb algebra in physics. Besides it treats howsome representations we have seen fall apart into indecomposable TLn-modules.

5.1 The spin chain model

First we give a short example of a model in statistical physics, called the Heisenberg XXZspin- 1

2chain model. The exposition is based on [15] and [27]. The second subsection makes

the connection to the Temperley-Lieb algebra and studies how the spin chain representationfrom section 3.2 decomposes in irreducibles.

5.1.1. The Heisenberg XXZ spin-12 chain model

We consider a quantum mechanical model in one dimension, called the XXZ model. Lookingthrough the eyes of a physicist, it describes a system of vertices on a one-dimensional latticewhere each vertex carries a so-called “quantum spin” which interacts with the neighbouringquantum spins. These quantum spins are given by elements in C2 and in the case of finitelymany vertices, a configuration of spins can be described as an element in

(C2)⊗n = C2 ⊗⋯⊗C2

(n copies of C2).The XXZ Hamiltonian is an operator

HXXZ = −1

2∑i∈Z

(σxi σxi+1 + σyi σyi+1 +∆σzi σ

zi+1),

where σx, σy and σz are the Pauli spin operators given in remark 3.14 and ∆ is a real number.The quantum spin chain specified by the hamiltonian HXXZ is called the XXZ model.

Among the physical problems of interest are the following:

• to diagonalize the Hamiltonian,

• to compute the matrix elements of a local operator with respect to its eigenvectors.

A local operator is a linear combination of products of finitely many spin operators. TheHamiltonian is an operator acting on the infinite tensor product ⋯ ⊗ C2 ⊗ C2 ⊗ ⋯, a systemof infinite number of degrees of freedom. In order to grasp its action, it is customary to startfrom a finite tensor product (C2)⊗n with cyclic boundary condition, e.g. σxi+n = σxi . Aftersolving this finite case, the limit n→∞ can be studied.

The connection to the spin chain module becomes apparent by the equality in (5.1). Recallfrom equation 3.7 that we may write

ζ(ei) =1

2(σxi σxi+1 + σyi σ

yi+1) −

1

4(q + q−1)(σzi σzi+1 − id) + 1

4(q − q−1)(σzi − σzi+1).

63

64 Chapter 5. Examples in statistical physics

When using cyclic boundary conditions (i.e. consider the indices modulo n, this will bestudied in the next chapter), it can be see that

n

∑i=1

ζ(ei) =1

2

n

∑i=1

(σxi σxi+1 + σyi σyi+1 −

1

2(q + q−1)(σzi σzi+1 − id)) + 1

4

n

∑i=1

((q − q−1)(σzi − σzi+1))

´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶=0

= 1

2

n

∑i=1

(σxi σxi+1 + σyi σyi+1 +∆σzi σ

zi+1 −∆ id ),

where ∆ = − 12(q + q−1). Hence we can write the Hamiltonian as

HXXZ = −n

∑i=1

(ζ(ei) +∆ id ). (5.1)

A more elaborate explanation on (exactly solved) models in statistical physics may be foundin [2].

5.1.2. Conjecture about the structure of (C2)⊗n

We have seen that we can write (C2)⊗n =n

⊕i=−ni even

En,i. Using equation (3.8) one can see that,

dimEn,−n = dimVn,0

dimEn,2−n = dimVn,1 + dimVn,0

dimEn,4−n = dimVn,2 + dimVn,1 + dimVn,0.

In general, when p ≥ 0 we have

dimEn,p−n = dimVn,p/2 + dimVn,p/2−1 + . . . + dimVn,0.

This insight leads to the following conjecture.

5.1 Conjecture. When TLn is semisimple,

(C2)⊗n =⌊n/2⌋

⊕i=0

(n − 2i + 1)Vn,i.

5.2 The dimer representation revisited

In section 3.3 we briefly introduced the Dimer representation. In this section we will study itsrelation to the Dimer model, which is a model used in statistical physics. The construction weto obtain a transfermatrix has appeared in [21], [22], [24] and more. The dimer representationand its connection to the dimer model were first given by Morin-Duchesne in 2015 [22].

In the second subsection, the structure of the Dimer representation is investigated. Whenn is odd, corollary 4.20 tells us that TLn(0) is semisimple, so it must be possible to write theDimer representation as the direct sum of standard modules.

5.2. The dimer representation revisited 65

5.2.1. Connection to the dimer model

The Dimer model is a model in statistical physics that was first introduced by Fowler andRushbrook in 1937 [9]. After its introduction it has been studied by Kasteleyn [17], Temperleyand Fister [8] and more. The theory of the dimer model is reviewed in lecture notes by Kenyon[19].

The model consists of a square grid in Z2 with m rows and n columns covered with dimers.A dimer is a vertical or horizontal tile that covers exactly two adjacent vertices. A dimerconfiguration is a distribution of the dimers such that each vertex is covered by precisely onedimer. It is possible to make periodic boundary conditions. We will identify the horizontallines at the top and bottom of the grid. The number of possible dimer configurations withoutidentifications on the boundary is studied in [17].

It is natural to assign to a covering a weight. Let γ, δ ∈ C×, let k be the number of horizontaldimers of a certain covering and let k′ = mn−2k

2be the number of vertical dimers. The the

weight of the covering is given by γk′δk. Since the total number of dimers is equal to 1

2nm

we know k′ = 12nm − k and the factor γk

′does not provide any extra information. Therefore,

set γ = 1, so that the weight of a covering is given by δk. The partition function of the dimermodel is given by the sum of the weights of all possible coverings,

Z(δ) = ∑coverings

δk.

Note that the total number of different (m × n)-dimer coverings is Z(1).A dimer configuration can be converted to a plane with uparrows and downarrows in

between two vertically neighbouring vertices, by placing an uparrow in the positions that arecovered by a vertical dimer and downarrows in the other positions. See for example figure5.1. Each row between two rows of vertices can then be associated with an element in (C2)⊗nby setting ↑= (1

0) ∈ C2 and ↓= (0

1) ∈ C2.

Figure 5.1: Example of a 6-by-8 dimer covering.

In [21], Lieb uses a transfer matrix to compute so called partition functions for the dimermodel. The transfer matrix builds all possible configurations of a row above a given row adimer configurations. If a given vertex (x, y) has two arrows pointing at it (one from aboveand one from below), then there must be a dimer from (x, y−1) to (x, y). If it has a downarrowabove and below it, then (x, y) must be pard of a horizontal dimer.

Let a row v ∈ (C2)⊗(n−1) be given. In order to find possible top rows, at first all arrows arereversed by applying V1 =∏n−1

i=1 σxi . Now, when there are two uparrows next to each other, the


two vertices may be covered by a horizontal dimer. The operator

V3 =n−1

∏i=2

(id+δσ−i σ−i+1) = exp(n−2

∑i=1

δσ−i σ−i+1)

constructs these horizontal dimers in all possible ways. The transfer matrix is the productV3V1,

T (δ) = exp(n−2

∑i=1

δσ−i σ−i+1)

n−1

∏i=1

σxi .

The procedure is illustrated in figure 5.2 and 5.3.

V1 V3 + δ + δ

Figure 5.2: Transfer matrix computing possible next rows.

V1 V3 + δ + δ

Figure 5.3: Computing possible next rows.

The way we encounter the dimer representation in the dimer model is as follows. Let theTLn(0)-representation τ be given by τ(1) = id and

τ(ei) = σ−i−1σ−i + σ−i σ−i+1 if i is odd

σ+i−1σ+i + σ+i σ+i+1 if i is even .

(One can easily check that this is a representation of TLn(0) on (C2)⊗(n−1).) Set U =∏n−1i=1,i odd σ

xi .

Then τ = U−1τU . The dimer model is connected to τ via

T 2(δ) =⌊n/2⌋

∏i=1

(id+δσ−i σ−i+1) ×n−1

∏i=1

(id+δσ−i σ−i+1)

=n−1

∏i=1i odd

(id+δσ−i−1σ−i )(id+δσ−i σ−i+1) ×

n−1

∏i=1i even

(id+δσ+i−1σ+i )(id+δσ+i σ+i+1)

=n−1

∏i=1i odd

(id+δσ−i−1σ−i + δσ−i σ−i+1) ×

n−1

∏i=1i even

(id+δσ+i−1σ+i + δσ+i σ+i+1)

=n−1

∏i=1i odd

( id+δτ(ei)) ×n−1

∏i=1i even

( id+δτ(ei)).

Hence the dimer representation τ is connected to the dimer model.

5.2.2. The structure of the Dimer representation

Following an article by Morin-Duchesne et al [22].

5.3. Fully and completely packed loop model 67

5.2 Proposition. Let n be odd and p ≥ 0. Then En−1,v decomposes into the direct sum

Ev ≅ Vn,2∣v∣+1 ⊕Vn,2∣v∣+5 ⊕ . . .⊕Vn,p′

with p′ = n or n − 2, depending on the value of n mod 4.

Proof. By proposition 3.23 hp,k is a homomorphism from Wn,(n−1−p−4k)/2 into En−1,p which isnonzero on Vn,(n−1−p−4k)/2. Let hp,k be the restriction of hp,k to Vn,(n−1−p−4k)/2. Since the lattermodule is irreducible, ker hp,k must be trivial. Since the Vn,p are non-isomorphic for differentp, the images of hp,k for 0 ≤ k ≤ ⌊n−1−p

4⌋ must be mutually disjoint. A dimension count now

proves the proposition. It is convenient to use the equality in equation (4.21) of [22].

A similar result holds for n odd and p < 0. The structure for even n is studied by Morin-Duchesne et al. in [22].

5.3 Fully and completely packed loop model

We finish this chapter with an application of the extended affine Temperley-Lieb algebra. Thefully packed loop model is an integrable system.

Consider a square grid in R2 consisting of the lines m × Z and Z ×m′ (with m,m′ ∈ Z).We can pick a finite subset by intersecting it with (− 1

2, n + 1

2) × (− 1

2, n + 1

2) (see figure figure

5.4, left). Mark every other outer edge and number them 1,2, . . . ,2n (figure 5.4, middle). AFully Packed Loop configuration is a colouring of paths such that each crossing is touchedby exactly one path (figure 5.4, right).

1 2

3

45

6

1 2

3

45

6

Figure 5.4: Fully packed loop model.

Looking at the endpoints of each path yields a matching of the numbers 1,2, . . . ,2n. Forexample, in figure 5.4 1 is matched to 2, 3 to 4 and 5 to 6.

The subject of interest is the probability of a matching occuring. If each configuration ofpaths is equally likely to appear, it is computed by counting the number of configurationswhich produce a given matching and dividing that number by the total number of configura-tions. As we can see, the calculation of this probability is a counting problem.

Now look at a cylinder with infinite height which is covered with tiles. Each tile is either(with probability 1

2) or (also with probability 1

2). Then we get a cylinder with paths on

it. See for example figure 5.5. This is called the Completely Packed Loop model. Numberingthe tiles on the bottom row and following the lines then yields a matching on 1,2, . . . ,2n.Clearly the probability of a certain matching is not a counting problem, since the cylinder hasinfinite height and there are infinitely many configurations.

The physicists Alexander Razumov and Yuri Stroganov conjectured in 2001 (in a paperpublished in 2004) that the probability of a certain matching given by a random CompletelyPacked Loop configuration is the same as the probability that the matching is given by a


Figure 5.5: A Completely Packed Loop configuretion.

random Fully Packed Loop configuration [25]. Special cases of the conjecture were proven,among others, by Francesco and Zinn-Justin [10], [11]. Finally, the conjecture was proven in2011 by Luigi Cantini and Andrea Sportiello in [4].

The proof uses the Temperley-Lieb algebra. The affine Temperley-Lieb algebra provides anaction on the set of matchings. This is not difficult to see: affine Temperley-Lieb diagrams canbe viewed as diagrams on a cylinder. Drawing this cylinder underneath a fully packed loopconfiguration yields a different matching, see also figure 5.6. The precise proof is given in [4]by Cantini and Sportiello and for a more elaborate version of the same proof we refer to abachelor’s thesis by Lewis Zwart [30].

Figure 5.6: The action of an aTL-diagram.

In conclusion, in this chapter we have seen that the Temperley-Lieb algebra does not onlyprovide a beautiful mathematical structure, but is useful outside mathematics. In particular,we have seen it appearing in statistical physical models.

Chapter 6

Representations of the affine Temperley-Lieb algebra

This chapter discusses some representation theory of the extended and reduced affine Temperley-Lieb algebra. The TLn-modules from chapter 3 are modified in order to become aTLn- andrTLn-modules.

The first section treats the analog of the link state-modules from section 3.1. Subsequently,the spin chain-module is modified to become an affine module. The intertwiner Ψ formsubsection 3.2.2 turns out to become an isomorphism of aTL-modules in subsection 6.2.1 andthe intertwiner Ω from subsection 3.2.3 is an inspiration to find an intertwiner Ω of rTL-modules in subsection 6.2.2.

The third section is concerned with generalising the dimer representation to the affine dimerrepresentation. Furthermore, the intertwiner Γ from proposition 3.21 is used as an inspirationto construct the map Γ which intertwiners affine link state-modules and the affine dimerrepresenation.

6.1 Link state-modules of aTL

In this section we let ourselves be inspired by representations of TLn(β) from section 3.1 tofind representations of the affine Temperley-Lieb algebra.

6.1.1. The matchmaker representation

During the discussion of the fully packed loop model in section 5.3 we have already en-countered an affine Temperley-Lieb representation (in disguise). This subsection makes thatrepresentation precise.

6.1 Definition. A matching on n points is a set of links on the points 1, . . . , n. A non-crossing matching is a matching such that the links do not cross. A perfect matching is a setof n/2 non-crossing links on n points (n is even).

Let n ∈ 2N and consider a circle with n enumerated vertices connected by n/2 non-crossingedges, such that each vertex is the endpoint of exactly one edge. This gives a perfect matchingon n points. Let Ln be the set of all possible perfect matchings on n points and write CLn forthe formal vector space over C with basis Ln.

Example. The set L6 consists of:

6.2 Lemma. Consider the vector space CL equipped with the map m ∶ aTLn(β)→ End(CLn) definedby

m(ei) ⋅m = β ⋅m if there is a link i↔ i + 1 in mm′ otherwise

m(u) ⋅m =m′′

69

70 Chapter 6. Representations of the affine Temperley-Lieb algebra

where m′ equals m, except the links connecting to i and i+1 are cut and rearranged such that i connectsto i + 1 and the two vertices that are left are connected, and m′′ arises from m by rotating m clockwiseby an angle of 2π/n.

The map m is an algebra homomorphism, hence it defines a representation of aTLn(β) on CLn,called the matchmaker representation.

Proof. We need to prove that m is a homomorphism.This can be seen easily by drawing figureslike in figure of the previous example.

In order to grasp the definition, we have a look at some examples.

Examples. Figure 6.1 gives a graphical representation of e1e4 and u acting on the first andthird diagram of figure 6.1.

= β =

= =

Figure 6.1: Actions of aTL6 on elements of L6.

6.1.2. The singles representation

As a generalisation of the matchmaker representation, we can allow defects to occur. For thedefinition we follow [23].

A (n, p)-singles diagram is a circle with n numbered vertices on the circle and a vertex inthe middle. It has n − 2p defects, which are represented by an edge from a vertex on thecircle to the middle vertex, and p links, which must be drawn in such a way that they do notcross the edges caused by the defects. Each diagrams gives a pairing of 2p vertices. Considerdiagrams modulo homotopy. Denote the set of (n, p)-singles diagrams by Sn,p.

Example. The set S5,2 is depicted in figure 6.2.

Example. The set S5,1 consists of the diagram in figure 6.3 and its rotations by 2kπ/5 radians.Figure 6.4 shows two diagrams that are not (5,1)-singles diagrams.

Remarks. (i) The set of diagrams Sn,s does not correspond with the canonical basis for Vn,pwith p = (n − 2)/2 (compare figure 6.3 and 3.1).

(ii) The set Ln (defined for even n) does not correspond to Sn,n/2, since the latter has avertex in the middle and an edge may pass that centre vertex in two ways. The numberof perfect matchings #Ln is easily seen to be equal to #Bn,n/2 = dn,n/2 from proposition2.5. On the other hand, #Sn,n/2 = ( n

n/2) as will be shown in lemma 6.5.

6.1. Link state-modules of aTL 71

n

1

Figure 6.2: The set S5,2.

1

2

Figure 6.3: The elements of S5,1.

Figure 6.4: Diagrams that are not in S5,1.

We are now ready to define the singles representation.

6.3 Definition. The singles map is the map µn,p ∶ rTLn(β,α) → End(CSn,p) given as follows.Let x ∈ rTLn(β,α) be a diagram and w ∈ Sn,p. Pick a ∈ C×. Then

(i) concatenate the diagrams as usual,(ii) remove contractible loops by multiplying with a factor β and non-contractible loops by

multiplying with a factor α,(iii) count for each of the n−2p defects how many steps it has moved clockwise (this number

might be negative), call these numbers ∆1, . . . ,∆n−2p, set ∆ = ∑n−2pi=1 ∆i and multiply the

result from (ii) with a factor a∆.

This procedure gives an element in CSn,p. We call a the twist parameter of µ. Linearlyextending this construction yields a map from rTLn(β,α) to End(CSn,p).

The following lemma states that µ defines a rTL-representation. The proof is straightfor-ward and will therefore be omitted.

6.4 Lemma. The map µn,p ∶ rTLn → End(CSn,p) is an algebra homomorphism, hence it defines arepresentation of rTLn on CSn,p, called the singles representation.

If no confusion can occur, we drop the subscript n, p from the µ. Putting a = 1 yields arepresentation without twist parameter. Before we continue the study, let us have a look at anexample to clarify the definition.


Example. Let

w = , and x = .

Then µ(u)w = a3w and

µ(x)w = a2 = a2β .

Remark. There is a natural way to orientate the links. Given a circle diagram, cut it open viathe line segment starting on the boundary between 1 and n and ending in the middle vertex(dashed in figure 6.5). Then bend the boundary to a vertical line, reflect the figure in the line(so that the diagram appears at the right of the line) and send middle vertex infinitely far tothe right. The vertex where a link is said to start is the vertex after which the link goes down(possibly trough the cut line back to the top). The procedure is illustrated in figure 6.5 withn = 7.

7

12

3

4

5 6

7

12

3

4

5 6

7

1

2

3

4

5

6

Figure 6.5: Orientation of links.

With this remark on the orientation of links, define for w ∈ Sn,p the set

ψ(w) = (j1, j′1), (j2, j′2), . . . , (jp, j′p), (6.1)

with the links numbered 1,2, . . . , p, where 1 ≤ jk < n denotes the beginning of the k-th linkand j′k with jk < j′k < n + jk the corresponding endpoint. This will be used in the constructionof the intertwiners Ω in definition 6.12 and Γ before proposition 6.16. However, in subsection6.2.1 we temporarily forget this discussion while defining the first intertwining operator ofthe chapter using a different procedure.

The construction the the remark gives rise to a way of counting #Sn,p.

6.5 Lemma. The number of diagrams in Sn,p is (np).

Proof. Using the construction from the remark above, it suffices to consider diagrams on avertical line where links cannot “go over” defects and links that go trough the dashed line at

6.2. Spin representations 73

the bottom reappear at the dashed line at the top. Encoding such a diagram by a vector of+ and −, where we put a − at position i in the vector if in the diagram a link closes at i anda + if a link starts or there is a defect, yields a bijective correspondence between diagrams inSn,p and vectors of length n with p entries equal to − and n − p entries equal to +. (Note that,given a sequence of + and − with #+ ≥ #−, one can uniquely reconstruct its correspondingdiagram, hence the bijection.) Such a vector can be made by choosing the p positions of the−’s, hence there are (n

p) such vectors, which proves the lemma.

We conclude the section with a remark on the relation between aTLn- and rTLn-modules.

6.6 Lemma. Let ρ be a representation of rTL on a vector space V . Then ρ′ ∶ aTL→ End(V ) given by

ρ(x)v = ρ([x])v,

where [x] ∈ rTLn denotes the equivalence class of x ∈ aTL, defines a representation of aTL on V .

6.2 Spin representations

In the first subsection we consider the extended affine Temperley-Lieb algebra (i.e. the algebrasatisfying relation (i) trough (v) from section 2.2) and see how the spin representation fromdefinition 3.13 can be extended to a representation of aTL, called the simple spin representa-tion. Moreover, we define an intertwiner Ψ ∶ CSn,n/2 → End ((C2)⊗n) for even n.

Subsequently, the condition that non-contractible loops may be removed by multiplyingwith a factor α is enforced on the algebra, so that the algebra rTL is obtained and α has to beincorporated in the spin representation. This will be called the reduced spin representation.With twist parameter a = 1 this representation lifts (cf. lemma 6.6) to the simple spin repre-sentation on aTL. Besides, an intertwiner between µ and the reduced spin representation isdefined by extending the construction from subsection 3.2.3.

6.2.1. Simple spin representation

Consider the extended affine Temperley-Lieb algebra and assume (in this subsection) that nis even. We extend the spin chain representation from subsection 3.2.1 to an aTL-module.Thereafter we give an intertwiner from the link state-module of perfect matchings to this newspin chain module, using the construction from subsection 3.2.2 and following [27].

We aim to define a representation ζ ∶ aTLn(β,1) → End((C2)⊗n) of the affine Temperley-Lieb algebra which is similar to the TLn-module from definition 3.13. Keeping in mind thisdefinition, it seems wise to set ζ(ei) = ζ(ei) for 1 ≤ i ≤ n − 1. The obvious way to constructζ(en) is as the same matrix but acting on the n-th and first component of the n-fold tensorproduct in (C2)⊗n (in that order). Thus

ζ(ei) =⎛⎜⎜⎜⎝

0 0 0 00 q 1 00 1 q−1 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

∈ End ((C2)⊗n)

where we think of the indices modulo n.


Since ζ has to be an affine Temperley-Lieb-module, the generators u and u−1 have to bedealt with. To this end, set

P =⎛⎜⎜⎜⎝

1 0 0 00 0 1 00 1 0 00 0 0 1

⎞⎟⎟⎟⎠

and Dγ = (γ+ 00 γ−

) ,

for some γ+, γ− ∈ C×. Then P is the matrix that sends (v⊗w) to (w⊗v) and Pi,j ∈ End ((C2)⊗n)permutes the i-th and the j-th component of a tensor. Define

ζ(u) = Pn−1,nPn−2,n−1⋯P1,2(Dγ)1, ζ(u−1) = (Dγ−1)1P1,2P2,3⋯Pn−1,n.

That isζ(u) ∶ (vε1 ⊗ vε2 ⊗⋯⊗ vεn)↦ (γε1vε2 ⊗⋯⊗ vεn ⊗ vε1).

The map ζ is called the simple spin representation of aTL on End ((C2)⊗n). The followinglemma states that it is indeed a representation.

6.7 Lemma. The map ζ satisfies the defining relations of aTLn form 2.17, hence ζ defines a represen-tation of aTLn.

Proof. This is a special case of the reduce spin representation given in definition 6.11 wherea = 1. Lemma 6.11 proves that it is a rTL-module, that is, it satisfies the relations (i) through(vi) given in section 2.2. In particular, it satisfies the first five, proving that ζ defines a aTL-representation.

Recall the the theory of orientations of a diagram (analogous to definition 3.15).

6.8 Definitions. Given the set Sn,n/2 of perfect matchings, orientate a link by choosing itsstarting point and endpoint and orientate an element of Sn.n/2 by choosing an orientation foreach link. Let Sn,n/2 be the set of all oriented perfect matchings on n points. Let Forg ∶ Sn,n/2 →Sn,n/2 be the function that forgets the orientation. For w ∈ Ln and j ∈ 1, . . . , n define

rj(L) ∶= + if the link at j is outgoing− if the link at j is incoming

and

or(w) ∶=# links from i to j with 1 ≤ j < i ≤ n −# links from i to j with 1 ≤ i < j ≤ n .

Example. Some possible orientations on an element of S6,3 (but not all, one can easily see that#Sn,n/2 = #Sn,n/2 ⋅ 2n/2).

View CSn,n/2 as aTL-module with respect to the matchmaker representation µn,n/2 withtwist parameter a = 1, and (C2)⊗n as aTL-module with respect to the simple spin represen-tation with γ = (γ+, γ−) = (q−1, q). The following proposition gives an intertwining operatorbetween the two representations.


6.9 Proposition. Define Ψ ∶ CLn → (C2)⊗n, linear, via

Ψ(w) ∶= ∑L∈Forg−1(L)

qor(w)/2 ⋅ vr1(w) ⊗⋯⊗ vrn(w).

Then Ψ is an intertwiner of aTL-modules and is injective for generic q.

Proof. Let w ∈ Sn,n/2, i ∈ 1, . . . , n. We need to check that

Ψ(µn,0(ei)w) = ζ(ei)Ψ(w) and Ψ(µn,0(u)w) = ζ(u)Ψ(w).

We subdivide this into the following cases:

(1) Ψ(µ(ei)w) = ζ(ei)Ψ(w) for 1 ≤ i < n and i is connected to i + 1 in L.

(2) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i+ 1 is connected to k and 1 ≤ j < i <i + 1 < k ≤ n.

(3) Ψ(µ(u)w) = ζ(u)Ψ(w).

(4) Ψ(µ(ei)w) = ζ(ei)Ψ(w) where i is connected to j and i + 1 is connected to k with j ≠ i + 1and k ≠ i.

Let us proceed. Case 1 and 2 are similar to the first two cases in the proof of proposition3.16. (For the second case, figure 6.6 is more suitable.)

k

n1

j

i i + 1

k

n1

j

i i + 1

Figure 6.6: Ad case 2.

Case 3. With the generator u. Let w′ = µ(u)w. Given an orientation w give w′ the canonicalorientation inherited from w. If rn(w) = + then or(w′) = or(w) + 2 and if rn(w) = − thenor(w′) = or(w) − 2.

Now compute

Ψ(µ(u)w) = ∑w∈Forg−1(w)

rn(w)=+

q−(or(w)+2)/2 ⋅ vr2(w) ⊗⋯⊗ vrn(w) ⊗ v+


rn(w)=−

q−(or(w)−2)/2 ⋅ vr2(w) ⊗⋯⊗ vrn(w) ⊗ v−,

and

ζ(u)Ψ(L) = ∑w∈Forg−1(w)

q−or(L)/2 ⋅ Pn−1,nPn−2,n−1⋯P1,2(Dγ)1(vr1(w) ⊗⋯⊗ vrn(w))


rn(w)=+

γ+q−or(L)/2 ⋅ vr2(w) ⊗⋯⊗ vrn(w) ⊗ v+


rn(w)=−

γ−q−or(L)/2 ⋅ v− ⊗ vr2(w) ⊗⋯⊗ vrn(w) ⊗ v−.


We conclude that Ψ(µ(u)w) = ζ(u)Ψ(w) for γ = (q−1, q).

Case 4. Given w ∈ L. Assume i is connected to j and i + 1 is connected to k. Recall thatei = u−(j−1)ei−(j−1)u

j−1 by definition 2.17 and note that u−(j−1)L is such that 1 is connected toi − (j − 1) and i + 1 − (j − 1) is connected to k − (j − 1) where i + 1 − (j − 1) < k ≤ n. (Then weare in the situation of case 2.) Compute

ζ(ei)Ψ(w) = ζ(u(j−1)ei−(j−1)u−(j−1))Ψ(w)

= ζ(u)j−1ζ(ei−(j−1))ζ(u−1)j−1Ψ(w)(case 2)= ζ(u)j−1ζ(ei−(j−1))Ψ(µ(u−1)j−1w)(case 3)= ζ(u)j−1Ψ(µ(ei−(j−1))µ(u−1)j−1w)(case 2)= Ψ(µ(u)j−1µ(ei−(j−1))µ(u−1)j−1w)= Ψ(µ(uj−1ei−(j−1)u

−(j−1))w)= Ψ(µ(ei)w).

Injectivity. Injectivity is proven in the same was as in proposition 3.16.

6.10 Corollary. The matchmaker representation and the simple spin representation on En,0 are iso-morphic aTL-modules.

Proof. Since Ψ gives an injective map, it suffices that the dimensions of the two representationsare the same. The space En,0 has a basis of pure tensors (vε1 ⊗⋯⊗ vεn) with n/2 of the vεi arev+ and the other half are v−. The number of such tensors is ( n

n/2). On the other hand, lemma

6.5 shows that dimCCSn,n/2 = ( nn/2

), thus proving the isomorphism.

6.2.2. Reduced spin representation

For the second time, we extend de spin representation from definition 3.13. Let a ∈ C× bethe twist parameter from definition 6.3. Besides, recall the notation β = q + q−1 = b2 + b−2 forq, b ∈ C× with q = b2. Define the affine spin representation as follows.

6.11 Lemma. Define the mapζ ∶ rTLn → End ((C2)⊗n)

by

ζ(ei) =⎛⎜⎜⎜⎝

0 0 0 00 b2 a2 00 a−2 b−2 00 0 0 0

⎞⎟⎟⎟⎠i,i+1

∈ End ((C2)⊗n)

acting on an element in (C2)⊗n and

ζ(u) ∶ (vε1 ⊗ vε2 ⊗⋯⊗ vεn)↦ (aSz

vε2 ⊗ vε2 ⊗⋯⊗ vεn ⊗ vε1),ζ(u−1) ∶ (vε1 ⊗ vε2 ⊗⋯⊗ vεn)↦ (a−S

z

vεn ⊗ vε1 ⊗⋯⊗ vεn−1).

Set α = an + an−. The map ζ satisfies the defining relations of rTLn from 2.17, hence ζ defines arepresentation of rTLn.


One can easily check that ζ(ei) may be written as

ζ(ei) = a−2σ−i σ+i+1 + a2σ+i σ

−i+1 − (b2 + b−2)σ+i σ−i σ+i+1σ

−i+1 + b2σ+i σ−i + b−2σ+i+1σ

−i+1, (6.2)

where σ−, σ+ are as in remark 3.14.So the simple spin representation is the special case of the extended affine spin represen-

tation where a = 1. Before we get too excited, let us prove that ζ indeed defines a rTL-representation on End ((C2)⊗n).

Despite the fact that the definition of the extended affine spin representation is based on theone defined by Morin-Duchesne in [23], our definition of the affine Temperley-Lieb algebradiffers slightly from his, which causes the proof to vary. Also, Morin-Duchesne chooses tocheck whether ζ satisfies the sixth defining relation of the algebra by a roundabout way,whereas we will check it by a direct computation.

Proof of lemma 6.11. By direct computation one can see that ζ(ei)2 = βζ(ei). Since ζ(ei) andζ(ej) do not act on the same components of a tensor when i ≠ j±1 mod n we have ζ(ei)ζ(ej) =ζ(ej)ζ(ei). The property ζ(ei)ζ(ei±1)ζ(ei) = ζ(ei) can be verified by a 8-by-8 matrix-check.

When i ≠ 1, the fourth relation, ζ(u)ζ(ei) = ζ(ei−1)ζ(u), can be computed straightforwardly,

ζ(u)ζ(ei)(vε1 ⊗⋯⊗ vεn)= ζ(u)(vε1 ⊗⋯⊗ ζ(e)(vεi ⊗ vεi+1)⊗⋯⊗ vεn)= aS

z

(vε2 ⊗⋯⊗ ζ(e)(vεi ⊗ vεi+1)⊗⋯⊗ vεn ⊗ vε1)= aS

z

ζ(ei−1)(vε2 ⊗⋯⊗ vεn ⊗ vε1)= ζ(ei−1)ζ(u)(vε1 ⊗⋯⊗ vεn).

When i = 1 a similar argument holds, but we have to think modulo n.The fifth relation is obvious. The vertices are moved up and down and the factors aS

z

anda−S

z

cancel out.Lastly, assume n is even and consider eodd = e1e3⋯en−1. We aim to show

ζ(eodd)ζ(u)ζ(eodd) = αζ(eodd). (6.3)

Let v = vε1 ⊗⋯ ⊗ vεn be a pure tensor in End ((C2)⊗n). If vεi ⊗ vεi+1 = v+ ⊗ v+ or v− ⊗ v− for iodd, then ζ(ei)(vεi ⊗ vεi+1) = 0 and the truth of identity in (6.3) is clear.

Now assume each pair εi, εi+1 (i odd) consists of one + and one −. The following observationis useful,

ζ(e)(v+ ⊗ v−) = b2(v+ ⊗ v−) + a−2(v− ⊗ v+)ζ(e)(v− ⊗ v+) = a2(v+ ⊗ v−) + b−2(v− ⊗ v+)

= b−2a2ζ(e)(v+ ⊗ v−).

Set v+ = (v+⊗v−⊗⋯⊗v+⊗v−) and v− = (v−⊗v+⊗⋯⊗v−⊗v+). Then ζ(eev)v− = b−nanζ(eev)v+.In general, if v is a pure tensor and m denotes the number of odd i where εi = −, thenζ(eev)v = b−2ma2mζ(eev)v+.


Consider the case v = v+. Then

ζ(eodd)ζ(u)ζ(eodd)v+ = ζ(eodd)ζ(u)((b2)n/2v+ + (a−2)n/2v− + rest)= ζ(eodd)(aS

z

bnv− + aSz

a−nv+ + ζ(u)(rest))= ζ(eodd)(aS

z

anb−nbnv+ + aSz

a−nv+ + ζ(u)(rest))= ζ(eodd)(aS

z

(an + a−n)v+) + ζ(eodd)ζ(u)(rest)´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶

=0

= αζ(eodd)v+.

Here we have used that aSz

commutes with the action of rTL and that Sz = 0.If v is a pure tensor such that for odd i (vεi ⊗ vεi+1) = (v+ ⊗ v−) or (vεi ⊗ vεi+1) = (v− ⊗ v+),

and it has m times a v− in an odd position, then

ζ(eev)ζ(u)ζ(eev)v = b−2ma2mζ(eev)ζ(u)ζ(eev)v+

= b−2ma2mαζ(eev)v+

= αζ(eev)v.

The analysis with eev instead of eodd is similar. This finishes the proof.

Putting α = an + a−n, we can define an intertwiner between the singles representation andthe spin representation as rTLn-modules. The construction is based on an article by Morin-Duchesne and Saint-Aubin [23]. This is an extension of proposition 3.17 from subsection 3.2.3.

6.12 Definition. Write β = b2+b−2 and choose the twist parameter a ∈ C× such that α = an+a−n.Let w ∈ Sn,p and set ψ(w) ∶= (j1, j′1), . . . , (jm, j′m), where 1 ≤ jk ≤ n denotes the beginning ofa link and j′k with jk < j′k ≤ N + jk the corresponding endpoint. Define the map Ωn,p ∶ CSn,p →(C2)⊗n by linearly extending

Ωn,p(w) ∶= ∏(j,j′)∈ψ(w)

(aj′−jbσ−j′ + a−(j

′−j)b−1σ−j )(v⊗n+ )

for w ∈ Sn,p.

6.13 Proposition. Set α = an + a−n and let x ∈ aTLn(β,α) and w ∈ Sn,p, then

Ωn,p(µn,p(x)w) = ζ(x)Ωn,p(w). (6.4)

Proof. It suffices to check the identity from equation (6.4) for x a generator of the affineTemperley-Lieb algebra and w ∈ Sn,p. This brings the proof of the proposition back to acase-by-case check with ten cases.

Note that σ+i σ−i (v⊗n+ ) = (v⊗n+ ), σ+i (v⊗n+ ) = 0 and σ−i σ

−i (v⊗n+ ) = 0. Denote by Yi(w) the product

∏(uσ−d′+u−1σ−d) over all (d, d′) ∈ ψ(w) that do not involve vertex i and i+1. This is well-definedbecause all σ−j commute with each other.

In the first eight cases we investigate how x = ei acts on different w ∈ Sn,p. These casescorrespond to figure 6.8. Case 9 and 10 deal with x = u and x = u−1.

Case 1. w has a defect at both i and i + 1. Then µn,s(ei)w ∈ CSn,s−2, hence the product iszero in CSn,s and Ω(µ(ei)w) = Ω(0) = 0. On the other hand, Ω(w) always has v+ in the i-thand (i + 1)-th component of the tensors, and since ζ(ei) acts as 0 on v+ ⊗ v+, ζ(ei)Ω(w) = 0.


Case 2. Vertex i is connected to i + 1 by a simple link in w. Now µ(ei)w = βw. On the otherhand, we can use the notation of ζ(ei) from (6.2) to find

ζ(ei)Ω(w) = ζ(ei)(abσ−i+1 + a−1b−1σ−i )Yi(w)(v⊗n+ )= (ab3σ+i σ−i σ−i+1 + a−1bσ−i σ

+i+1σ

−i+1 + a−1b−3σ−i σ

+i+1σ

−i+1 + ab−1σ+i σ

−i σ

−i+1)Yi(w)(v⊗n+ )

= (ab3σ−i+1 + a−1bσ−i + a−1b−3σ−i + ab−1σ−i+1)Yi(w)(v⊗n+ )= (b2 + b−2)(abσ−i+1 + a−1b−1σ−i )Yi(w)(v⊗n+ )= (b2 + b−2)Ω(w).

Case 3. w has a link from j to i and a defect at i+ 1. In this case µ(ei)w has a defect at j anda link from i to i + 1 and is multiplied by a factor ai+1−j . Write µ(ei)w = ai+1−jw′. Compute

ζ(ei)Ω(w) = ζ(ei)(ai−jbσ−i + a−(i−j)b−1σ−j )Yi(w)(v⊗n+ )= (ai−jb−1σ−i σ

+i+1σ

−i+1 + ai−j+2bσ+i σ

−i σ

−i+1)Yi(w)(v⊗n+ )

= (ai−jb−1σ−i + ai−j+2bσ−i+1)Yi(w)(v⊗n+ )= ai+1−j(abσ−i+1 + a−1b−1σ−i )Yi(w)(v⊗n+ )= ai+1−jΩ(w′).

Case 4. w has a link from i + 1 to k and a defect at i. The proof of this case is similar to thatof case 3.

Case 5. w has a link from k to i + 1 and from j to i. The proof of this case is similar to thatof case 7.

Case 6. w has a link from j to i and from i + 1 to k. The proof of this case is similar to thatof case 7.

Case 7. w has a link from i to j and from i + 1 to k and both i and i + 1 are the beginning ofa link. The link state µ(ei)w has a link from i to i + 1 and from k to j. We have

ζ(ei)Ω(w) = ζ(ei)(aj−ibσ−j + a−(j−i)b−1σ−i )(ak−(i+1)bσ−k + a−(k−i−1)b−1σ−i+1)Yi(w)(v⊗n+ )= ζ(ei)(aj−iak−i−1b2σ−kσ

−j + aj−ia−(k−i−1)σ−i+1σ

−j

+ a−(j−i)ak−i−1σ−i σ−k + a−(j−i)a−(k−i−1)b−2σ−i σ

−i+1)Yi(w)(v⊗n+ )

= (aj−k+1b2σ−i+1σ−j + ak−j+1σ−i+1σ

−k + aj−k−1σ−i σ

−k + ak−j−1b−2σ−i σ

−k)Yi(w)(v⊗n+ )

= (abσ−i+1 + a−1b−1σ−i )(aj−kbσ−j + a−(j−k)b−1σ−k)Yi(w)(v⊗n+ )= Ω(µ(ei)w).

Case 8. w has a link from i + 1 to n + i. In this case µ(ei)w results in a simple link from i toi + 1 and multiplication by a factor α (see figure 6.7). On the other hand, we can compute

ζ(ei)Ω(w)= ζ(ei)(an−1bσ−i + ab−(n−1)σ−i+1)Yi(w)(v⊗n+ )= (an−1b−1σ−i σ

+i+1σ

−i+1 + an+1bσ+i σ

−i σ

−i+1 + a−(n−1)bσ+i σ

−i σ

−i+1 + a−(n−1)b−1σ−i σ

+i+1σ

−i+1)Yi(w)(v⊗n+ )

= (an + a−n)(abσ−i+1 + a−1b−1σ−i )Yi(w)(v⊗n+ )= Ω(µ(ei)w).


w =i

i + 1µ(ei)w =

i

i + 1 = α ⋅i

i + 1

Figure 6.7: Emergence of a loop.

Now consider x = u.

Case 9. Let w by any diagram in Sn,p. Note ζ(u)(aj′−jbσ−j′ + a−(j′−j)b−1σ−j ) = a−2(aj′−jbσ−j′ +

a−(j′−j)b−1σ−j )ζ(u) and ζ(u)(v⊗n+ ) = an(v⊗n+ ). Finally note n − 2p equals the number of defects

and µ(u) moves each defect one position clockwise, hence harvesting a factor an−2p. Compute

ζ(u)Ω(w) = ζ(u)( ∏(j,j′)∈ψ(w)

(aj′−jbσ−j′ + a−(j

′−j)b−1σ−j ))(v⊗n+ )

= a−2∣ψ(w)∣( ∏(j,j′)∈ψ(w)

(aj′−jbσ−j′−1 + a−(j

′−j)b−1σ−j−1))ζ(u)(v⊗n+ )

= ana−2p( ∏(j,j′)∈ψ(w)

(aj′−jbσ−j′−1 + a−(j

′−j)b−1σ−j−1))(v⊗n+ )

= Ω(µ(u)w).

Case 10. The proof that ζ(u−1)Ω(w) = Ω(µ(u−1)w) is similar to that of case 9.

The question whether Ω is an isomorphism is studied in detail in [23].

6.3 Affine dimer representation

It appears to be possible to extend the idea of the dimer representation on TLn(0) from[22], treated in section 3.3, to the extended affine Temperley-Lieb algebra with β = 0 and thereduced affine Temperley-Lieb algebra with β = 0 and α = 2, which is done in this section.First, the affine Dimer representation is defined and shown to be a aTLn(0)- and rTLn(0,2)-module (lemma 6.15). Thereafter, an operator Γp is constructed and proposition 6.16 provesthat Γp intertwines µn,p and τn−2p as rTLn(0,2)-modules.

6.14 Definition. The affine dimer map τ ∶ aTLn(0) → End ((C2)⊗n) is the linear map τ givenby

τ(1) = id

τ(ei) = σ−i−1σ+i + σ+i σ−i+1

τ(u) ∶ (vε1 ⊗ vε2 ⊗⋯⊗ vεn)↦ (vε2 ⊗⋯⊗ vεn ⊗ vε1)τ(u−1) ∶ (vε1 ⊗ vε2 ⊗⋯⊗ vεn)↦ (vεn ⊗ vε1 ⊗⋯⊗ vεn−1).

By checking that τ satisfies the conditions in definition 2.17 it can be seen that it defines arepresentation, which is called the affine dimer representation. The treatment of (i) through

6.3. Affine dimer representation 81

(iii) is analogue to the proof of lemma 3.20 and (v) is immediate from the construction. For(iv), we show τ(u)σ−i−1σ

+i = σ−i−2σ

+i−1τ(u) and τ(u)σ+i σ−i+1 = σ+i−1σ

−i τ(u). Let v = vε1 ⊗⋯⊗ vεn ∈

(C2)⊗n. If either vεi−1 = v− or vεi = v+ we have τ(u)σ−i−1σ+i = 0 = σ−i−2σ

+i−1τ(u). If not, then

vεi−1 ⊗ vεi = v+ ⊗ v− and

τ(u)σ−i−1σ+i (v) = τ(u)(vε1 ⊗⋯⊗ vεi−2 ⊗ v− ⊗ v+ ⊗ vεi+1 ⊗⋯⊗ vεn)

= (vε2 ⊗⋯⊗ vεi−2 ⊗ v− ⊗ v+ ⊗ vεi+1 ⊗⋯⊗ vεn ⊗ vε1)= σ−i−2σ

+i−1(⊗vε2 ⊗⋯⊗ vεi−2 ⊗ v− ⊗ v+ ⊗ vεi+1 ⊗⋯⊗ vεn ⊗ vε1)

= σ−i−2σ+i−1τ(u)(v).

Note that in the second and third line v− and v+ are in the (i − 1)-th and i-th position ofthe tensor product. In a similar way it can be seen that τ(u)σ+i σ−i+1 = σ+i−1σ

−i τ(u), ultimately

proving τ(u)τ(ei) = τ(ei−1)τ(u).Moreover, τ turns out to be a rTL-module for β = 0 and α = 2.

6.15 Lemma. The map τ ∶ rTLn(0,2) → End ((C2)⊗n) defined as in definition 6.14 is a representa-tion of rTLn(0,2) on (C2)⊗n.

Proof. Prior to this lemma it was shown that τ satisfies relations (i) through (v) from thedefinition of the affine Temperley-Lieb algebra given in section 2.2, so only relation (vi) (for neven) is still open. Consider first the case

τ(eev)τ(u)τ(eev) = ατ(eev). (6.5)

Note

τ(eev) = τ(e2)τ(e4)τ(e6)⋯τ(en)= (σ−1σ+2 + σ+2σ−3 )(σ−3σ+4 + σ+4σ−5 )(σ−5σ−6 + σ−6σ−7 )⋯(σ−n−1σ

+n + σ+nσ−1 )

= 2σ−1σ+2σ

−3σ

+4⋯σ−n−1σ

+n,

where the last equality holds since only the product over all the first terms and the productover all second terms is nonzero. If w ∈ (C2)⊗n is such that τ(eev)w = 0 then (vi) is triviallysatisfied. The kernel of τ(eev) equals

ker τ(eev) = span(vε1 ⊗⋯⊗ vεn) ∣ (ε1, ε2, . . . , εn−1, εn) ≠ (+,−, . . . ,+,−).Thus the only pure tensor in the canonical basis of (C2)⊗n is v+ = (v+ ⊗ v− ⊗⋯⊗ v+ ⊗ v−). Butthen

v+ 2v− 2v+ 4v−τ(eev) τ(u) τ(eev)

2τ(eev)

where v− = (v−⊗ v+⊗⋯⊗ v−⊗ v+). It follows that the identity in equation (6.5) holds for α = 2.The cases τ(eev)τ(u−1)τ(eev) = 2τ(eev) and τ(eodd)τ(u±1)τ(eodd) = 2τ(eodd) can be proven ina similar way.

Let τn−2p be the restriction of τ to En,p. Inspired by the non-affine case, define the mapΓp ∶ CSn,p → En,n−2p as follows. Let w ∈ Sn,p be a link state and set ψ(w) as described afterequation (6.1). Set σn+m ≡ σm and define the map Γp by

Γp(w) ∶= ∏(j,j′)∈ψ(w)

(σ−j−1 + σ−j′)(v⊗n+ )

when w ∈ Sn,p.


6.16 Proposition. Let α = 2. The map Γp intertwines the rTLn(0,2)-modules µn,p and τn−2p.

Proof. As usual by now, we need to prove for x ∈ rTLn(0,2) and w ∈ Sn,p the identity

Γp(µn,p(x)w) = τn−2p(x)Γp(w) (6.6)

and it suffices to check this for x = 1, x = ei and x = u±1, and w is a link state. We have 10

cases. The first eight cases consists of x = ei and w is as in figure 6.8. The last two cases arex = u and x = u−1 with w ∈ Sn,p arbitrary.

w =i + 1

i

i + 1i

i + 1i

j

k

i + 1i

k

i + 1i

j

i + 1i

j

k

j

k

i + 1i

i + 1i


Case 1 – 7. The treatment of these seven cases is analogous to the proof of proposition 3.21.

Case 8. We haveΓp(µp(ei)w) = αYi(w)(σ−i−1 + σ−i+1)(v⊗n+ ).

On the other hand, one can compute

τn−2p(ei)Γp(w) = (σ−i−1σ+i + σ+i σ−i+1)Yi(w)(σ−i + σ−i )(v⊗n+ )

= 2(σ−i−1σ+i + σ+i σ−i+1)σ−i Yi(w)(v⊗n+ )

= 2(σ−i−1 + σ−i+1)Yi(w)(v⊗n+ ).

Thus the intertwining property holds when α = 2.

Case 9. When x = u, simply compute,

Γp(µ(u)w) = ∏(j,j′)∈ψ(µ(u)w)

(σ−j−1 + σ−j′)(v⊗n+ )

= ∏(j,j′)∈ψ(w)

(σ−j + σ−j′+1)(v⊗n+ )

= τn−2p(u) ∏(j,j′)∈ψ(w)

(σ−j−1 + σ−j )(v⊗n+ )

= τn−2p(u)Γp(w).

Case 10. One can prove Γp(µ(u−1)w) = τn−2p(u−1)Γp(w) in a similar way as case 9.

This completes the proof that Γp is an intertwining operator.

Chapter 7

Conclusion

The thesis started with various versions of the Temperley-Lieb algebra in chapter 2, whichcould be seen as quotients of the Hecke algebra and the group algebra of the braid group.Furthermore, a new central element, Jn, of the Temperley-Lieb algebra was defined.

Subsequently, in chapter 3, we have given an overview of well-known representations ofthe Temperley-Lieb algebra and connections between them. This resulted in the diagram infigure 7.1.

Wn,p Vn−1,p−1↑ Vn+1,p↓

En−1,n−1−2p Vn,p ⊕Vn,p−1

En,p ⊕ En,p−1

Γp β = 0∼

∼

Φ

∼ (n,p) non-critical

Ωn,p⊕Ωn,p−1

Figure 7.1: Module identities for generic q.

The structure of the Temperley-Lieb algebra has been studied in detail in chapter 4. Herewe followed an article by Ridout and Saint-Aubin [26] except for one mayor difference: weused the newly found central element Jn instead of Fn, the central element used in the article.This asked for a slight modification of some proofs. The Temperley-Lieb algebra turned outto be semisimple for generic q. In the non-semisimple case, the principal indecomposablemodules and a complete set of irreducibles were constructed.

In order to give purpose to the study of the Temperley-Lieb algebra, chapter 5 showed con-nections between the Temperley-Lieb algebra and several statistical physical models: the spinchain model, the dimer model and the fully and completely packed loop model. Althoughthe author thinks no justification is necessary to practise mathematics, for those who disagreethis chapter might justify the study of the Temperley-Lieb algebra.

The thesis concluded in chapter 6 with a study of the representation theory of the extendedand reduced affine Temperley-Lieb algebra. Many representations and intertwining operatorsthat were defined for the non-affine Temperley-Lieb algebra served as an inspiration to obtainaTL- and rTL-modules and module homomorphisms. Most notably, the dimer representationwas generalised to the reduced affine Temperley-Lieb algebra and a homomorphism betweena diagram representation called the singles representation of rTLn on CSn,p and the affinedimer representation was given. To the author’s best knowledge, this has not appeared inliterature before.

In the thesis, there are two obvious starting points for further research. The first one is atconjecture 5.1. Although this looks like an easy statement, the author has not found it inliterature.

Second, the dimer representation was defined only for β = 0 and immediately raises thequestion whether a similar construction is possible for β ≠ 0. The same holds for the affinedimer representation. Besides, concerning the affine dimer representation, both the rTL-

83

84 Chapter 7. Conclusion

representation and intertwiner Γ work precisely when α = 2. So α = 2 seems to be a naturalchoice. Where this naturality comes from is not directly clear and could be a starting pointfor further research.

Chapter 8

Populaire samenvatting

Rekenen met diagrammen

Op de basisschool en middelbare school leren we om te rekenen met getallen. Bijvoorbeeldoptellen en vermenigvuldigen, 4+6 = 10 en 3×7 = 21. Maar het is ook mogelijk om te rekenenmet andere objecten. In deze scriptie rekenen we bijvoorbeeld met diagrammen. Een diagramziet er als volgt uit:

(i) Teken twee verticale lijnen.(ii) Teken op elke lijn n punten (bijvoorbeeld n = 4).

(iii) Teken lijnen tussen de punten zodat elk punt verbonden is met een lijn, de lijnen elkaarniet snijden en zodat de lijnen tussen de twee verticale lijnen lopen.

Figuur 8.1 geeft een voorbeeld van de constructie. Er zijn natuurlijk veel meer diagrammenmogelijk. (Probeer er zelf een paar te tekenen!)

Figure 8.1: Constructie van een diagram.

Het optellen en aftrekken van de diagrammen gaat hetzelfde als met variabelen. Stel datx en y verschillende diagrammen zijn, dan is x + x = 2x, maar x + y kunnen we niet korterschrijven omdat x en y niet hetzelfde zijn. We tellen de diagrammen dus alleen op als ze gelijkzijn. Hetzelfde geldt voor aftrekken, 3x − 2x = x, maar 3x − 2y blijft gewoon 3x − 2y.

Vermenigvuldigen van de diagrammen is leuker. Het gaat als volgt:

(i) Zet de twee diagrammen achter elkaar.(ii) Verwijder de middelste verticale lijn met punten.

(iii) Verwijder lijnen die niet met een van de buitenste lijnen verbonden zijn.

× = = = =

Figure 8.2: Vermenigvuldigen van diagrammen.

Als x en y diagrammen zijn, dan is x × y niet altijd hetzelfde als y × x. Dat kunnen webijvoorbeeld zien door figuur 8.2 en 8.3 te vergelijken. (Probeer zelf twee diagrammen x en yte verzinnen en x × y en y × x uit te rekenen.)

85

86 Chapter 8. Populaire samenvatting

× = = =

Figure 8.3: Vermenigvuldigen van diagrammen.

De verzameling van (sommen van) diagrammen wordt de Temperley-Lieb algbera ge-noemd. We noteren de verzameling (met de bijbehorende structuur van optellen, aftrekken envermenigvuldigen) met TLn, waar de kleine n staat voor het aantal punten op elke verticalelijn. Het zijn de “getallen” waar we in deze scriptie mee rekenen. In de wiskunde noemen wedeze structuur van optellen, aftrekken en vermenigvuldigen een “algebra.” Merk op dat wein onze algebra niet kunnen delen.

Halve diagrammen

Door een diagram verticaal door midden te snijden krijgen we een half diagram. Het kangebeuren dat er losse eindjes overblijven, die noemen we defecten. Figuur 8.4 geeft eenvoorbeeld.

Figure 8.4: Constructie van een half diagram.

Halve diagrammen kunnen we op dezelfde manier optellen en aftrekken als gewone dia-grammen. We noemen de verzameling van (sommen van) halve diagrammen het link mod-uul en noteren hem met Mn, waarbij de kleine n weer het aantal punten op een verticale lijnaangeeft.

Stel dat x een diagram is en v een half diagram. Dan kunnen we de twee combineren eneen nieuw half diagram krijgen. Dit gaat als volgt:

(i) Zet x en v achter elkaar.(ii) Verwijder de rechter verticale lijn met punten.

(iii) Verwijder de lijnen die niet met de linker lijn zijn verbonden.

Deze procedure heet de actie van x op v en we schrijven die met x ⋅ v. Figuur 8.5 geeft eenvoorbeeld van deze procedure.

Kies x nu vast. Dan kunnen we voor elk half diagram v in Mn een nieuw half diagramvinden door de actie x ⋅ v van x op v uit te rekenen. Het diagram x geeft dus een formuleom van een half diagram een nieuw half diagram te maken. We noemen de formule fx en ergeldt dus

fx(v) = x ⋅ v.

87

⋅ = = =

Figure 8.5: De actie van een diagram op een half diagram.

Het is niet moeilijk te geloven dat elk element x in TLn zo’n formule oplevert. We kunnen nuopnieuw een formule vinden, die we ρ noemen. In ρ kan je een element uit de Temperley-Liebalgebra stoppen en er komt een formule zoals hierboven uit. Dus elk element in TLn leidt toteen formule fx van Mn naar Mn. Voor x in TLn schrijven we

ρ(x) = fx.

In de wiskunde zeggen we nu dat ρ een representatie is van TLn op Mn. Representatieszijn wiskundige hulpmiddelen om een algebra te bestuderen.

Verder in deze scriptie

In deze scriptie staan representaties van de Temperley-Lieb algebra centraal. Er zijn namelijknog veel meer representaties te bedenken, maar die zijn vaak een stuk moeilijker. Het blijkt inveel gevallen mogeljk te zijn om “bouwstenen” te vinden waar alle representaties van gemaaktzijn.

Ook bekijken we variaties op te Temperley-Lieb algebra en representaties daarvan. Totslot laten we zien hoe de Temperley-Lieb algebra en haar representaties gebruikt worden in(theoretische) natuurkunde.

88 Chapter 8. Populaire samenvatting

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[23] A. Morin-Duchesne and Y. Saint-Aubin, A homomorphism between link and XXZ modules over the periodic temperley-lieb algebra, Journal of Physics A: Mathematical and Theoretical 46(28) (2013), 34–.

[24] J. Rasmussen and P. Ruelle, Refined conformal spectra in the dimer model, Journal of Statistical Mechanics: Theoryand Experiment 2012(10) (2012).

[25] A.V. Razumov and Yu. G. Stroganov, Combinatorial nature of ground state vector of O(1) loop model, Theoretical andMathematical Physics 138 (2004), 333 – 337.

[26] D. Ridout and Y. Saint-Aubin, Standard modules, induction and the structure of the Temperley-Lieb algebra (2012).

[27] J. V. Stokman, GQT course: Quantum integrable systems, Lecture notes, University of Amsterdam, 2014, https://staff.fnwi.uva.nl/j.v.stokman/QIntSystemGQTnotes.pdf.

89

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90 Bibliography

[28] H. Temperley and E. Lieb, Relations Between the ‘Percolation’ and ‘Colouring’ Problem and Other Graph-TheoreticProblems Associated with Regular Plane Lattices: Some Exact Results for the ‘Percolation’ Problem, Proceeds of theRoyal Society of London 322 (1971), 251 – 280.

[29] B. W. Westbury, The representation theory of the Temperley-Lieb algebra, Mathematische Zeitschrift 219 (1995), 539 –566.

[30] L. Zwart, Proof of the Razumov-Stroganov conjecture, Bachelor’s thesis, University of Amsterdam, 2014, https://esc.fnwi.uva.nl/thesis/apart/math/thesis.php?page=math.

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https://esc.fnwi.uva.nl/thesis/apart/math/thesis.php?page=math

Chapter A

Miscellaneous

A.1 The central element Jn

A.1 Proposition. The element Jn acts on Vn,p as the identity times gn,p = (−1)n(q(n−2p+1) +q−(n−2p+1)).

We remark that gn,p = (−1)nfn,p, where fn,p denotes the action of Fn on Vn,p given inproposition A.2 of [26].

Proof. Any central element acts as a scalar times the identity on Vn,p by proposition 4.6. Sowe may compute the action of Jn on any (n, p)-link state. Let zp denote the element in Vn,pwith simple links at 1, 3, . . . , 2p − 1. Then by using the Reidermeister moves we find

Jnzp =

1

2

2p − 12p

=

Figure A.1: Using Reidermeister 2 on Jnzp.

Now let us concentrate on the lower n − 2p levels. We focus on the diagram left in figure ...and evaluate the top right tile

Jnzp =

2p + 12p + 2

n − 1n

= c ⋅ + c−1 ⋅

Figure A.2: The last n − 2p levels.

Focus on the first diagram on the right-hand side of figure A.2 and consider the crossingbelow the one we just changed (the right one on level 2p + 2). Filling in the tile results inan extra link (use Reidermeister 2 again) thus giving coefficient 0 in Vn,p. Hence only the tile

results in a nonzero coefficient. We obtain the equality in figure A.3. We can recycle theargument to fill in all of the right column.

91

92 Appendix A. Miscellaneous

c ⋅ = c2 ⋅ = cn−2p ⋅

Figure A.3: Expanding the right column of the diagram.

Now have a look at the left column. In order to not close defects (making the diagram 0),there is one possibility for the top left tile, and, using the same argument, for all tiles exceptthe lowest one in the column. This is depicted in figure A.4

cn−2p+1 ⋅ = c2(n−2p)−1 ⋅ = c2(n−2p) ⋅ + c2(n−2p)−2 ⋅

Figure A.4: Expanding the left column of the diagram.

We end with the factor βc2n−4p + c2n−4p−2. Recall c2 = −q and c2 + c−2 = −β to find

βc2(n−2p) + c2(n−2p)−2 = (−c2 − c−2)c2(n−2p) + c2(n−2p)−2 = −c2 ⋅ c2(n−2p) = (−1)nqn−2p+1.

In a similar way, we may treat the right diagram on the right-hand side of figure A.1 tofind a factor (−1)nq−(n−2p+1). Thus the total factor equals gn,p = (−1)n(qn−2p+1 + q−(n−2p+1)), asdesired.

A.2 Preliminary representation theory

A.2 Definitions. Recall that for an associative algebra A a complex vector space V is called arepresentation of A or A-module if there exists a homomorphism of algebras ρ ∶ A→ End(V ),i.e. a linear map ρ with ρ(1) = id, the identity map on V , and ρ(ab) = ρ(a)ρ(b). The elementρ(a)(v) ∈ V is often denoted ρ(a)v or av for short.

A subspace W ⊆ V is called a subrepresentation if it is invariant under the operators ρ(a) ∶V → V . A representation is said to be irreducible if it has no proper subrepresentations andindecomposible if it is not isomorphic to the direct sum of two nonzero subrepresentations.A semisimple or completely reducible representation is a representation that is the directsum of irreducible representations.

A homomorphism or intertwining operator (or intertwiner for short) of representations isa linear map φ between A-modules V and W such that aφ(v) = φ(av) for all a ∈ A,v ∈ V . It iscalled an isomorphism if in addition φ is bijective.

A.2. Preliminary representation theory 93

A.3 Theorem (Wedderburn). Let A be a complex, finite-dimensional, semisimple, associative algebrawith a complete set of irreducibles L1, . . .Lr. Then

A ≅r

⊕i=1

(dimLi)Li.

On the other hand,

A.4 Proposition. Let A be a complex, finite-dimensional, associative algebra with complete set ofirreducibles L1, . . .Lr. View A as module over itself and suppose

A ≅r

⊕i=1

(dimLi)Li.

Then A is semisimple.

A.5 Theorem. Let A be a complex, finite-dimensional, associative algebra. Let Pi be the set ofindecomposables and Li be the set of irreducibles. There is a bijective correspondence between the set.Let Li bet the irreducible quotient of Pi. If r is the common cardinality of these sets, then the regularrepresentation decomposes as

A ≅r

⊕i=0

(dimLi)Pi.

A.6 Proposition. (Frobenius reciprocity). Let B ⊂ A be two finite-dimensional associative algebrasover C. Let M be a B-module and N be an A-module. Then, the following isomorphism between vectorspaces of module homomorphisms holds:

HomA(M↑,N) ≅ HomB(M,N↓).

A.7 Proposition. Suppose M1,M2,M3 are B-modules and

0 M1 M2 M3 0

is a short exact sequence. Let A be an algebra such that B ⊂ A and Mi↑ = A⊗B Mi, then

M1↑ M2↑ M3↑ 0

is exact.

an introduction to the representation theory of temperley ...the primary goal of this thesis is to...

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